The Database of Integer Sequences, Part 5
Part of the On-Line Encyclopedia of Integer Sequences
This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.
For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )
Seis.html: Welcome
index.html: Lookup
indexfr.html: Francais
demo1.html: Demos
Sindx.html: Index
WebCam.html: WebCam
Submit.html: Contribute new sequence or comment
eishelp1.html: Internal format
eishelp2.html: Beautified format
transforms.html: Transforms
Spuzzle.html: Puzzles
Shot.html: Hot
classic.html: Classics
ol.html: Superseeker
JIS/index.html: Journal of Integer Sequences
pages.html: More pages
Maintained by: N. J. A. Sloane (njas@research.att.com),
home page: www.research.att.com/~njas/
(start)
%I A025660
%S A025660 0,1,0,2,1,0,3,2,1,0,4,3,2,1,0,5,4,3,2,1,0,6,5,4,3,2,1,0,7,6,5,4,3,2,1,0,
%T A025660 8,7,6,5,4,3,2,1,0,9,8,7,6,5,4,3,2,1,0,10,9,8,7,6,5,4,3,2,1,0,11,10,9,8,
%U A025660 7,6,5,4,3,2,1,0,12,11,10,9,8,7,6,5,4,3,2,1,13,0,12,11,10,9,8,7,6,5,4,3
%N A025660 Exponent of 6 (value of i) in n-th number of form 6^i*7^j.
%Y A025660 Adjacent sequences: A025657 A025658 A025659 this_sequence A025661 A025662 A025663
%Y A025660 Sequence in context: A025669 A025676 A025683 this_sequence A025677 A025651 A025670
%K A025660 nonn
%O A025660 1,4
%A A025660 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025677
%S A025677 0,1,0,2,1,0,3,2,1,0,4,3,2,1,0,5,4,3,2,1,0,6,5,4,3,2,1,0,7,6,5,4,3,2,1,0,
%T A025677 8,7,6,5,4,3,2,1,0,9,8,7,6,5,4,3,2,1,0,10,9,8,7,6,5,4,3,2,1,11,0,10,9,8,
%U A025677 7,6,5,4,3,2,12,1,11,0,10,9,8,7,6,5,4,3,13,2,12,1,11,0,10,9,8,7,6,5,4,14
%N A025677 Exponent of 8 (value of i) in n-th number of form 8^i*10^j.
%Y A025677 Adjacent sequences: A025674 A025675 A025676 this_sequence A025678 A025679 A025680
%Y A025677 Sequence in context: A025676 A025683 A025660 this_sequence A025651 A025670 A122200
%K A025677 nonn
%O A025677 1,4
%A A025677 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025651
%S A025651 0,1,0,2,1,0,3,2,1,0,4,3,2,1,0,5,4,3,2,1,0,6,5,4,3,2,1,0,7,6,5,4,3,2,1,0,
%T A025651 8,7,6,5,4,3,2,1,0,9,8,7,6,5,4,3,2,1,10,0,9,8,7,6,5,4,3,2,11,1,10,0,9,8,
%U A025651 7,6,5,4,3,12,2,11,1,10,0,9,8,7,6,5,4,13,3,12,2,11,1,10,0,9,8,7,6,5,14,4
%N A025651 Exponent of 5 (value of i) in n-th number of form 5^i*6^j.
%Y A025651 Adjacent sequences: A025648 A025649 A025650 this_sequence A025652 A025653 A025654
%Y A025651 Sequence in context: A025683 A025660 A025677 this_sequence A025670 A122200 A025646
%K A025651 nonn
%O A025651 1,4
%A A025651 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025670
%S A025670 0,1,0,2,1,0,3,2,1,0,4,3,2,1,0,5,4,3,2,1,0,6,5,4,3,2,1,0,7,6,5,4,3,2,1,0,
%T A025670 8,7,6,5,4,3,2,1,9,0,8,7,6,5,4,3,2,10,1,9,0,8,7,6,5,4,3,11,2,10,1,9,0,8,
%U A025670 7,6,5,4,12,3,11,2,10,1,9,0,8,7,6,5,13,4,12,3,11,2,10,1,9,0,8,7,6,14,5
%N A025670 Exponent of 7 (value of i) in n-th number of form 7^i*9^j.
%Y A025670 Adjacent sequences: A025667 A025668 A025669 this_sequence A025671 A025672 A025673
%Y A025670 Sequence in context: A025660 A025677 A025651 this_sequence A122200 A025646 A025661
%K A025670 nonn
%O A025670 1,4
%A A025670 David W. Wilson (davidwwilson(AT)comcast.net)
%I A122200
%S A122200 0,1,0,2,1,0,3,2,1,0,4,3,2,1,0,5,4,3,2,1,0,6,5,4,3,2,1,0,7,6,5,4,3,2,
%T A122200 1,0,8,8,6,5,4,3,2,1,0,9,7,7,6,5,4,3,2,1,0,10,9,8,7,6,5,4,3,2,1,0,11,
%U A122200 10,9,8,7,6,5,4,3,2,1,0,12,11,10,9,8,7,6,5,4,3,2,1,0,13,13,11,10,9,8
%N A122200 Signature permutations of RIBS-transformations of non-recursive Catalan automorphisms in table A089840.
%C A122200 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "RIBS".
%C A122200 In this recursion scheme the given automorphism is applied to all (toplevel) subtrees of the Catalan structure, when it is interpreted as a general tree. Permutations in this table form a countable group, which is isomorphic with the group in A089840. (The RIBS transformation gives the group isomorphism.)
%C A122200 Furthermore, row n of this table is also found as the row A123694(n) in tables A122203 and A122204. If the count of fixed points of the automorphism A089840[n] is given by sequence f, then the count of fixed points of the automorphism A089840[A123694(n)] is given by CONV(f,A000108) (where CONV stands for convolution), and the count of fixed points of the automorphism A122200[n] by INVERT(RIGHT(f)).
%C A122200 The associated Scheme-procedures RIBS and !RIBS can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism.
%C A122200 Comment from Antti Karttunen, May 11 2008: This sequence agrees with A025581 in its initial terms, but then diverges from it.
%D A122200 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122200 Index entries for signature-permutations of Catalan automorphisms
%o A122200 (Scheme) (define (RIBS foo) (lambda (s) (map foo s)))
%o A122200 (define (!RIBS foo!) (letrec ((bar! (lambda (s) (cond ((pair? s) (foo! (car s)) (bar! (cdr s)))) s))) bar!))
%Y A122200 Row 0 (identity permutation): A001477, row 1: A122282. See also tables A089840, A122201-A122204, A122283-A122284, A122285-A122288, A122289-A122290.
%Y A122200 Adjacent sequences: A122197 A122198 A122199 this_sequence A122201 A122202 A122203
%Y A122200 Sequence in context: A025677 A025651 A025670 this_sequence A025646 A025661 A025671
%K A122200 nonn,tabl,new
%O A122200 0,4
%A A122200 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006
%I A025646
%S A025646 0,1,0,2,1,0,3,2,1,0,4,3,2,1,0,5,4,3,2,1,0,6,5,4,3,2,1,0,7,6,5,4,3,2,1,8,
%T A025646 0,7,6,5,4,3,2,9,1,8,0,7,6,5,4,3,10,2,9,1,8,0,7,6,5,4,11,3,10,2,9,1,8,0,
%U A025646 7,6,5,12,4,11,3,10,2,9,1,8,0,7,6,13,5,12,4,11,3,10,2,9,1,8,0,7,14,6,13
%N A025646 Exponent of 4 (value of i) in n-th number of form 4^i*5^j.
%Y A025646 Differs from A025661 at a(1881).
%Y A025646 Adjacent sequences: A025643 A025644 A025645 this_sequence A025647 A025648 A025649
%Y A025646 Sequence in context: A025651 A025670 A122200 this_sequence A025661 A025671 A025652
%K A025646 nonn
%O A025646 1,4
%A A025646 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025661
%S A025661 0,1,0,2,1,0,3,2,1,0,4,3,2,1,0,5,4,3,2,1,0,6,5,4,3,2,1,0,7,6,5,4,3,2,1,8,
%T A025661 0,7,6,5,4,3,2,9,1,8,0,7,6,5,4,3,10,2,9,1,8,0,7,6,5,4,11,3,10,2,9,1,8,0,
%U A025661 7,6,5,12,4,11,3,10,2,9,1,8,0,7,6,13,5,12,4,11,3,10,2,9,1,8,0,7,14,6,13
%N A025661 Exponent of 6 (value of i) in n-th number of form 6^i*8^j.
%Y A025661 Differs from A025646 at a(1881).
%Y A025661 Adjacent sequences: A025658 A025659 A025660 this_sequence A025662 A025663 A025664
%Y A025661 Sequence in context: A025670 A122200 A025646 this_sequence A025671 A025652 A025662
%K A025661 nonn
%O A025661 1,4
%A A025661 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025671
%S A025671 0,1,0,2,1,0,3,2,1,0,4,3,2,1,0,5,4,3,2,1,0,6,5,4,3,2,1,7,0,6,5,4,3,2,8,1,
%T A025671 7,0,6,5,4,3,9,2,8,1,7,0,6,5,4,10,3,9,2,8,1,7,0,6,5,11,4,10,3,9,2,8,1,7,
%U A025671 0,6,12,5,11,4,10,3,9,2,8,1,7,13,0,6,12,5,11,4,10,3,9,2,8,14,1,7,13,0,6
%N A025671 Exponent of 7 (value of i) in n-th number of form 7^i*10^j.
%Y A025671 Adjacent sequences: A025668 A025669 A025670 this_sequence A025672 A025673 A025674
%Y A025671 Sequence in context: A122200 A025646 A025661 this_sequence A025652 A025662 A025640
%K A025671 nonn
%O A025671 1,4
%A A025671 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025652
%S A025652 0,1,0,2,1,0,3,2,1,0,4,3,2,1,0,5,4,3,2,1,6,0,5,4,3,2,7,1,6,0,5,4,3,8,2,7,
%T A025652 1,6,0,5,4,9,3,8,2,7,1,6,0,5,10,4,9,3,8,2,7,1,6,0,11,5,10,4,9,3,8,2,7,1,
%U A025652 12,6,0,11,5,10,4,9,3,8,2,13,7,1,12,6,0,11,5,10,4,9,3,14,8,2,13,7,1,12,6
%N A025652 Exponent of 5 (value of i) in n-th number of form 5^i*7^j.
%Y A025652 Adjacent sequences: A025649 A025650 A025651 this_sequence A025653 A025654 A025655
%Y A025652 Sequence in context: A025646 A025661 A025671 this_sequence A025662 A025640 A025663
%K A025652 nonn
%O A025652 1,4
%A A025652 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025662
%S A025662 0,1,0,2,1,0,3,2,1,0,4,3,2,1,0,5,4,3,2,1,6,0,5,4,3,2,7,1,6,0,5,4,3,8,2,7,
%T A025662 1,6,0,5,4,9,3,8,2,7,1,6,0,5,10,4,9,3,8,2,7,1,6,11,0,5,10,4,9,3,8,2,7,12,
%U A025662 1,6,11,0,5,10,4,9,3,8,13,2,7,12,1,6,11,0,5,10,4,9,14,3,8,13,2,7,12,1,6
%N A025662 Exponent of 6 (value of i) in n-th number of form 6^i*9^j.
%Y A025662 Adjacent sequences: A025659 A025660 A025661 this_sequence A025663 A025664 A025665
%Y A025662 Sequence in context: A025661 A025671 A025652 this_sequence A025640 A025663 A025647
%K A025662 nonn
%O A025662 1,4
%A A025662 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025640
%S A025640 0,1,0,2,1,0,3,2,1,0,4,3,2,1,5,0,4,3,2,6,1,5,0,4,3,7,2,6,1,5,0,4,8,3,7,2,
%T A025640 6,1,5,0,9,4,8,3,7,2,6,1,10,5,0,9,4,8,3,7,2,11,6,1,10,5,0,9,4,8,3,12,7,2,
%U A025640 11,6,1,10,5,0,9,4,13,8,3,12,7,2,11,6,1,10,5,0,14,9,4,13,8,3,12,7,2,11,6
%N A025640 Exponent of 3 (value of i) in n-th number of form 3^i*4^j.
%Y A025640 Adjacent sequences: A025637 A025638 A025639 this_sequence A025641 A025642 A025643
%Y A025640 Sequence in context: A025671 A025652 A025662 this_sequence A025663 A025647 A025653
%K A025640 nonn
%O A025640 1,4
%A A025640 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025663
%S A025663 0,1,0,2,1,0,3,2,1,0,4,3,2,1,5,0,4,3,2,6,1,5,0,4,3,7,2,6,1,5,0,4,8,3,7,2,
%T A025663 6,1,5,0,9,4,8,3,7,2,6,1,10,5,0,9,4,8,3,7,2,11,6,1,10,5,0,9,4,8,3,12,7,2,
%U A025663 11,6,1,10,5,0,9,4,13,8,3,12,7,2,11,6,1,10,5,14,0,9,4,13,8,3,12,7,2,11,6
%N A025663 Exponent of 6 (value of i) in n-th number of form 6^i*10^j.
%Y A025663 Adjacent sequences: A025660 A025661 A025662 this_sequence A025664 A025665 A025666
%Y A025663 Sequence in context: A025652 A025662 A025640 this_sequence A025647 A025653 A131103
%K A025663 nonn
%O A025663 1,4
%A A025663 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025647
%S A025647 0,1,0,2,1,0,3,2,1,0,4,3,2,1,5,0,4,3,2,6,1,5,0,4,3,7,2,6,1,5,0,4,8,3,7,2,
%T A025647 6,1,5,9,0,4,8,3,7,2,6,10,1,5,9,0,4,8,3,7,11,2,6,10,1,5,9,0,4,8,12,3,7,
%U A025647 11,2,6,10,1,5,9,0,13,4,8,12,3,7,11,2,6,10,1,14,5,9,0,13,4,8,12,3,7,11,2
%N A025647 Exponent of 4 (value of i) in n-th number of form 4^i*6^j.
%Y A025647 Differs from A025653 at a(2805).
%Y A025647 Adjacent sequences: A025644 A025645 A025646 this_sequence A025648 A025649 A025650
%Y A025647 Sequence in context: A025662 A025640 A025663 this_sequence A025653 A131103 A096652
%K A025647 nonn
%O A025647 1,4
%A A025647 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025653
%S A025653 0,1,0,2,1,0,3,2,1,0,4,3,2,1,5,0,4,3,2,6,1,5,0,4,3,7,2,6,1,5,0,4,8,3,7,2,
%T A025653 6,1,5,9,0,4,8,3,7,2,6,10,1,5,9,0,4,8,3,7,11,2,6,10,1,5,9,0,4,8,12,3,7,
%U A025653 11,2,6,10,1,5,9,0,13,4,8,12,3,7,11,2,6,10,1,14,5,9,0,13,4,8,12,3,7,11,2
%N A025653 Exponent of 5 (value of i) in n-th number of form 5^i*8^j.
%Y A025653 Differs from A025647 at a(2805).
%Y A025653 Adjacent sequences: A025650 A025651 A025652 this_sequence A025654 A025655 A025656
%Y A025653 Sequence in context: A025640 A025663 A025647 this_sequence A131103 A096652 A025654
%K A025653 nonn
%O A025653 1,4
%A A025653 David W. Wilson (davidwwilson(AT)comcast.net)
%I A131103
%S A131103 0,0,1,0,2,1,0,3,2,1,0,4,3,8,1,0,5,4,21,22,1,0,6,5,40,63,52,1,0,7,6,65,
%T A131103 124,243,114,1,0,8,7,96,205,664,969,240,1,0,9,8,133,306,1405,3196,3657,
%U A131103 494,1,0,10,9,176,427,2556,7425,15712,12987,1004,1,0,11,10,225,568,4207
%N A131103 Rectangular array read by antidiagonals: a(n, k) is the number of ways to put k labeled objects into n labeled boxes so that there are no boxes with exactly one object (n, k >= 1).
%C A131103 Problem suggested by Brandon Zeidler. Columns four and five are A000567 and A051874. Second row is A130102.
%F A131103 a(n, k) = sum_{j=1..min(floor(k/2), n)} A008299(k, j)*n!/(n-j)!.
%e A131103 Array begins:
%e A131103 0 1 1 1 1 1 1
%e A131103 0 2 2 8 22 52 114
%e A131103 0 3 3 21 63 243 969
%Y A131103 Cf. A131104, A131105, A131106, A131107.
%Y A131103 Adjacent sequences: A131100 A131101 A131102 this_sequence A131104 A131105 A131106
%Y A131103 Sequence in context: A025663 A025647 A025653 this_sequence A096652 A025654 A025648
%K A131103 easy,nonn,tabl
%O A131103 1,5
%A A131103 David Wasserman (dwasserm(AT)earthlink.net), Jun 14 2007, Jun 15 2007
%I A096652
%S A096652 1,0,1,0,2,1,0,3,2,1,0,5,5,2,1,0,7,7,7,2,1,0,11,16,9,9,2,1,0,15,15,31,11,11,2,1,0,22,
%T A096652 59,4,54,13,13,2,1,0,30,109,313,72,87,15,15,2,1,0,42,1314,1922,1122,225,132,17,17,
%U A096652 2,1,0,56,11804,19468,9671,3087,509,191,19,19,2,1,0,77,133957,217176,110734,32581
%V A096652 1,0,1,0,2,1,0,3,2,1,0,5,5,2,1,0,7,7,7,2,1,0,11,16,9,9,2,1,0,15,15,31,11,11,2,1,0,22,
%W A096652 59,-4,54,13,13,2,1,0,30,-109,313,-72,87,15,15,2,1,0,42,1314,-1922,1122,-225,132,17,17,
%X A096652 2,1,0,56,-11804,19468,-9671,3087,-509,191,19,19,2,1,0,77,133957,-217176,110734,-32581
%N A096652 Lower triangular matrix T, read by rows, such that the row sums of T^n form the (2n)-dimensional partition numbers.
%C A096652 Row sums of T form A000219 (planar partitions); row sums of T^2 form A000334(4-D); row sums of T^3 form A000416(6-D).
%F A096652 Matrix square of triangle A096651.
%e A096652 Triangle T begins:
%e A096652 {1},
%e A096652 {0,1},
%e A096652 {0,2,1},
%e A096652 {0,3,2,1},
%e A096652 {0,5,5,2,1},
%e A096652 {0,7,7,7,2,1},
%e A096652 {0,11,16,9,9,2,1},
%e A096652 {0,15,15,31,11,11,2,1},
%e A096652 {0,22,59,-4,54,13,13,2,1},
%e A096652 {0,30,-109,313,-72,87,15,15,2,1},
%e A096652 {0,42,1314,-1922,1122,-225,132,17,17,2,1},
%e A096652 {0,56,-11804,19468,-9671,3087,-509,191,19,19,2,1},
%e A096652 {0,77,133957,-217176,110734,-32581,7137,-980,266,21,21,2,1},
%e A096652 {0,101,-1728760,2809257,-1426436,422732,-87714,14601,-1704,359,23,23,2,1},...
%e A096652 Row sums are: {1,1,3,6,13,24,48,86,160,282,500,859,...} (A000219).
%e A096652 T^2 begins:
%e A096652 {1},
%e A096652 {0,1},
%e A096652 {0,4,1},
%e A096652 {0,10,4,1},
%e A096652 {0,26,14,4,1},
%e A096652 {0,59,38,18,4,1},
%e A096652 {0,140,109,50,22,4,1},
%e A096652 {0,307,256,179,62,26,4,1},
%e A096652 {0,684,709,370,273,74,30,4,1},
%e A096652 {0,1464,1240,1683,438,395,86,34,4,1},...
%e A096652 with row sums: {1,1,5,15,45,120,326,835,2145,5345,...} (A000334).
%Y A096652 Cf. A096651, A000219, A000334, A000416.
%Y A096652 Adjacent sequences: A096649 A096650 A096651 this_sequence A096653 A096654 A096655
%Y A096652 Sequence in context: A025647 A025653 A131103 this_sequence A025654 A025648 A025655
%K A096652 sign,tabl
%O A096652 0,5
%A A096652 Paul D. Hanna (pauldhanna(AT)juno.com), Jul 06 2004
%I A025654
%S A025654 0,1,0,2,1,0,3,2,1,4,0,3,2,5,1,4,0,3,6,2,5,1,4,0,7,3,6,2,5,1,8,4,0,7,3,6,
%T A025654 2,9,5,1,8,4,0,7,3,10,6,2,9,5,1,8,4,0,11,7,3,10,6,2,9,5,1,12,8,4,0,11,7,
%U A025654 3,10,6,2,13,9,5,1,12,8,4,0,11,7,3,14,10,6,2,13,9,5,1,12,8,4,15,0,11,7,3
%N A025654 Exponent of 5 (value of i) in n-th number of form 5^i*9^j.
%Y A025654 Adjacent sequences: A025651 A025652 A025653 this_sequence A025655 A025656 A025657
%Y A025654 Sequence in context: A025653 A131103 A096652 this_sequence A025648 A025655 A022336
%K A025654 nonn
%O A025654 1,4
%A A025654 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025648
%S A025648 0,1,0,2,1,0,3,2,1,4,0,3,2,5,1,4,0,3,6,2,5,1,4,7,0,3,6,2,5,8,1,4,7,0,3,6,
%T A025648 9,2,5,8,1,4,7,0,10,3,6,9,2,5,8,1,11,4,7,0,10,3,6,9,2,12,5,8,1,11,4,7,0,
%U A025648 10,3,13,6,9,2,12,5,8,1,11,4,14,7,0,10,3,13,6,9,2,12,5,15,8,1,11,4,14,7
%N A025648 Exponent of 4 (value of i) in n-th number of form 4^i*7^j.
%Y A025648 Adjacent sequences: A025645 A025646 A025647 this_sequence A025649 A025650 A025651
%Y A025648 Sequence in context: A131103 A096652 A025654 this_sequence A025655 A022336 A019586
%K A025648 nonn
%O A025648 1,4
%A A025648 David W. Wilson (davidwwilson(AT)comcast.net)
%I A025655
%S A025655 0,1,0,2,1,0,3,2,1,4,0,3,2,5,1,4,0,3,6,2,5,1,4,7,0,3,6,2,5,8,1,4,7,0,3,6,
%T A025655 9,2,5,8,1,4,7,10,0,3,6,9,2,5,8,11,1,4,7,10,0,3,6,9,12,2,5,8,11,1,4,7,10,
%U A025655 0,13,3,6,9,12,2,5,8,11,1,14,4,7,10,0,13,3,6,9,12,2,15,5,8,11,1,14,4,7
%N A025655 Exponent of 5 (value of i) in n-th number of form 5^i*10^j.
%Y A025655 Adjacent sequences: A025652 A025653 A025654 this_sequence A025656 A025657 A025658
%Y A025655 Sequence in context: A096652 A025654 A025648 this_sequence A022336 A019586 A063942
%K A025655 nonn
%O A025655 1,4
%A A025655 David W. Wilson (davidwwilson(AT)comcast.net)
%I A022336
%S A022336 0,1,0,2,1,0,3,2,1,4,0,3,2,5,1,4,0,3,6,2,5,1,4,7,0,3,6,2,5,8,1,4,7,0,3,6,
%T A022336 9,2,5,8,1,4,7,10,0,3,6,9,2,5,8,11,1,4,7,10,0,3,6,9,12,2,5,8,11,1,4,7,10,
%U A022336 13,0,3,6,9,12,2,5,8,11,14,1,4,7,10,13,0,3,6,9,12,15,2,5,8,11,14,1,4,7
%N A022336 Exponent of 3 (value of i) in n-th number of form 3^i*5^j.
%C A022336 a(n) = A007949(A003593(n)) = A112754(n) - A022337(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 18 2005
%Y A022336 Cf. A112761, A022329, A022337.
%Y A022336 Adjacent sequences: A022333 A022334 A022335 this_sequence A022337 A022338 A022339
%Y A022336 Sequence in context: A025654 A025648 A025655 this_sequence A019586 A063942 A106384
%K A022336 nonn
%O A022336 1,4
%A A022336 Clark Kimberling (ck6(AT)evansville.edu)
%I A019586
%S A019586 0,0,0,1,0,2,1,0,3,2,1,4,0,5,3,2,6,1,7,4,0,8,5,3,9,2,10,6,1,11,7,4,12,0,
%T A019586 13,8,5,14,3,15,9,2,16,10,6,17,1,18,11,7,19,4,20,12,0,21,13,8,22,5,23,
%U A019586 14,3,24,15,9,25,2,26,16,10,27,6,28,17,1,29,18,11,30,7,31,19,4,32,20,12
%N A019586 Vertical para-Fibonacci sequence: takes value i on later (i.e. b_j, j >= 2) terms of i-th Fibonacci sequence defined by b_0 = i, b_1 = [ tau(i+1) ].
%C A019586 Gives number of row in Wythoff array that contains n. - Casey Mongoven (cm(AT)caseymongoven.com), Sep 10 2005
%D A019586 J. H. Conway, personal communication.
%H A019586 N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
%H A019586 N. J. A. Sloane, Classic Sequences
%F A019586 Says which row of Wythoff array (starting row count at 0) contains n.
%F A019586 If delete first occurrence of 0, 1, 2, 3, ... the sequence is unchanged.
%Y A019586 Equals A003603(n) - 1.
%Y A019586 Cf. Wythoff array: A035513.
%Y A019586 Adjacent sequences: A019583 A019584 A019585 this_sequence A019587 A019588 A019589
%Y A019586 Sequence in context: A025648 A025655 A022336 this_sequence A063942 A106384 A094314
%K A019586 nonn,nice,easy,eigen
%O A019586 1,6
%A A019586 njas and J. H. Conway (conway(AT)math.princeton.edu)
%E A019586 Casey Mongoven (CaseyBach(AT)aol.com) reports that where the sequence reads 15,9,2,16,10,6,29,1,30,11,7,19..., the 29 and 30 should be 17 and 18.
%E A019586 More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
%I A063942
%S A063942 1,0,1,2,1,0,3,2,1,4,3,2,5,4,3,6,5,4,7,6,5,8,7,6,9,8,7,10,9,8,11,10,9,12,11,10,13,12,
%T A063942 11,14,13,12,15,14,13,16,15,14,17,16,15,18,17,16,19,18,17,20,19,18,21,20,19,22,21,20,
%U A063942 23,22,21,24,23,22,25,24,23,26,25,24,27,26,25,28,27,26,29,28,27,30,29,28,31,30,29,32
%V A063942 1,0,-1,2,1,0,3,2,1,4,3,2,5,4,3,6,5,4,7,6,5,8,7,6,9,8,7,10,9,8,11,10,9,12,11,10,13,12,
%W A063942 11,14,13,12,15,14,13,16,15,14,17,16,15,18,17,16,19,18,17,20,19,18,21,20,19,22,21,20,
%X A063942 23,22,21,24,23,22,25,24,23,26,25,24,27,26,25,28,27,26,29,28,27,30,29,28,31,30,29,32
%N A063942 Follow n by n-1 and n-2.
%o A063942 (PARI) a(n) = (n\3)-(n%3)+1; j=[]; for(n=0,200,j=concat(j,a(n))); j
%Y A063942 Cf. A028242.
%Y A063942 Adjacent sequences: A063939 A063940 A063941 this_sequence A063943 A063944 A063945
%Y A063942 Sequence in context: A025655 A022336 A019586 this_sequence A106384 A094314 A036864
%K A063942 easy,sign
%O A063942 0,4
%A A063942 Jason Earls (jcearls(AT)cableone.net), Sep 01 2001
%I A106384
%S A106384 2,1,0,3,2,1,5,1,4,6,8,9,24,14,23,32,40
%N A106384 Imaginary parts of numbers defined in A106383.
%Y A106384 Cf. A106383.
%Y A106384 Adjacent sequences: A106381 A106382 A106383 this_sequence A106385 A106386 A106387
%Y A106384 Sequence in context: A022336 A019586 A063942 this_sequence A094314 A036864 A058604
%K A106384 nonn
%O A106384 1,1
%A A106384 Sven Simon (sven-h.simon(AT)t-online.de), Apr 30 2005
%I A094314
%S A094314 1,0,1,0,0,2,1,0,3,2,2,8,4,8,2,13,30,40,20,15,2,80,192,210,152,60,
%T A094314 24,2,579,1344,1477,994,469,140,35,2,4738,10800,11672,7888,3660,
%U A094314 1232,280,48,2
%N A094314 Triangle read by rows: T(n,k) = number of ways of seating n couples around a circular table so that exactly k married couples are adjacent (0 <= k <= n).
%C A094314 The men and women alternate. The n-th row sums to n!.
%D A094314 Anthony C. Robin, Circular Wife Swapping, The Mathematical Gazette, November 2006.
%Y A094314 Essentially a mirror image of A058087, which has much more information. Diagonals give A000179, A000425, A000033, A000159, A000181, etc.
%Y A094314 Adjacent sequences: A094311 A094312 A094313 this_sequence A094315 A094316 A094317
%Y A094314 Sequence in context: A019586 A063942 A106384 this_sequence A036864 A058604 A072661
%K A094314 nonn,tabl
%O A094314 0,6
%A A094314 njas, based on a suggestion from Anthony Robin (anthony_robin(AT)hotmail.com), Jun 02 2004
%I A036864
%S A036864 0,0,0,0,0,1,0,0,1,0,2,1,0,3,2,3,2,2,6,8,6,5,5,13,23,13,11,14,25,55,31,
%T A036864 25,33,53,115,75,54,73,114,228,165,123,151,243,440
%N A036864 Number of partitions satisfying either one of the two conditions cn(0,5) = cn(1,5) <= cn(3,5) <= cn(2,5) = cn(4,5) or cn(2,5) = cn(4,5) <= cn(3,5) <= cn(0,5) = cn(1,5).
%C A036864 For a given partition cn(i,n) means the number of its parts equal to i modulo n.
%C A036864 Short: (0 = 1 <= 3 <= 2 = 4) or (2 = 4 <= 3 <= 0 = 1).
%Y A036864 Adjacent sequences: A036861 A036862 A036863 this_sequence A036865 A036866 A036867
%Y A036864 Sequence in context: A063942 A106384 A094314 this_sequence A058604 A072661 A103432
%K A036864 nonn,more
%O A036864 1,11
%A A036864 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%I A058604
%S A058604 1,2,1,0,3,2,3,4,0
%V A058604 1,2,-1,0,3,-2,3,4,0
%N A058604 McKay-Thompson series of class 27d for Monster.
%D A058604 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%e A058604 T27d = 1/q + 2*q^2 - q^5 + 3*q^11 - 2*q^14 + 3*q^17 + 4*q^20 + 4*q^26 + ...
%Y A058604 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%Y A058604 Adjacent sequences: A058601 A058602 A058603 this_sequence A058605 A058606 A058607
%Y A058604 Sequence in context: A106384 A094314 A036864 this_sequence A072661 A103432 A103448
%K A058604 sign
%O A058604 -1,2
%A A058604 njas, Nov 27, 2000
%I A072661
%S A072661 0,1,0,2,1,0,3,2,4,1,0,2,3,0,5,4,6,1,4,3,2,8,1,0,2,5,0,7,6,4,5,4,6,3,0,
%T A072661 9,8,10,1,8,3,2,12,1,0,2,7,8,5,4,6,5,4,7,6,16,1,0,2,9,0,11,10,4,9,8,10,
%U A072661 3,0,13,12,14,1,12,3,2,8,9,8,10,5,8,7,6,12,5,4,6,7,0,17,16,18,1,16,3,2
%N A072661 Composition of the A059905 and A048679, i.e. a(n) = A059905(A048679(n)).
%Y A072661 The other side of this projection is A072662. Used to construct the permutations A072657 and A072659.
%Y A072661 Adjacent sequences: A072658 A072659 A072660 this_sequence A072662 A072663 A072664
%Y A072661 Sequence in context: A094314 A036864 A058604 this_sequence A103432 A103448 A078803
%K A072661 nonn
%O A072661 0,4
%A A072661 Antti Karttunen Jun 02 2002
%I A103432
%S A103432 1,2,1,0,3,2,4,1,5,2,6,1,5,4,0,7,2,6,5,8,3,8,5,9,4,10,1,10,3,8,7,0,11,
%T A103432 4,10,7,11,6,13,2,10,9,12,7,14,1,15,15,13,8,15,4,16,1,13,10,14,9,16,5,
%U A103432 17,2,13,12,14,11,16,9,18,5,17,8,0,18,7,17,10,19,6,20,1,20,3,15,14,17
%N A103432 Subsequence of the Gaussian primes, where only Gaussian primes a+bi with a>0, b>=0 are listed. Ordered by the norm N(a+bi)=sqrt(a^2+b^2) and the size of the real part, when the norms are equal. The sequence gives the imaginary parts. See A103431 for the real parts.
%C A103432 Detailed description in A103431.
%Y A103432 Adjacent sequences: A103429 A103430 A103431 this_sequence A103433 A103434 A103435
%Y A103432 Sequence in context: A036864 A058604 A072661 this_sequence A103448 A078803 A130403
%K A103432 nonn
%O A103432 1,2
%A A103432 Sven Simon (sven-h.simon(AT)t-online.de), Feb 05 2005; corrected Feb 20 2005 and again on Aug 06 2006
%I A103448
%S A103448 1,2,1,0,3,2,6,4,1,2,6,4,0,6,8,6,2,2,2,4,4,10,4,8,0,4,8,2,4,0,2,4,2,4,0,4,2,4,10,
%T A103448 4,0,8,6,2,4,4,8,2,2,2,2,4,6,2,0,6,2,2,2,6,0,6,4,8,2,4,2,0,0,8,4,2,2,4,2,0,2,14,
%U A103448 10,2,2,2,4,2,4,2,0,8,4,2,2,2,6,0,6,14,2,0,2,2,2,4,0,2,2
%V A103448 1,2,1,0,3,2,6,4,1,2,6,4,0,-6,8,6,2,-2,2,-4,4,10,4,8,0,4,8,2,4,0,-2,-4,2,4,0,4,2,-4,10,
%W A103448 4,0,-8,6,-2,4,-4,8,2,2,2,2,4,6,2,0,6,2,2,2,-6,0,6,4,8,2,4,2,0,0,8,-4,-2,2,4,2,0,-2,14,
%X A103448 10,-2,2,2,4,2,4,-2,0,8,4,2,2,-2,6,0,-6,14,2,0,2,2,2,4,0,2,-2
%N A103448 a(n)=sum(mobius(binom(n,k)), k=0..n).
%C A103448 Row sums of A103447.
%F A103448 a(n)=sum(mobius(binom(n, k)), k=0..n).
%e A103448 a(4)=3 because mobius(1)+mobius(4)+mobius(6)+mobius(4)+mobius(1)=1+0+1+0+1=3.
%Y A103448 Cf. A103447, A103449.
%Y A103448 Adjacent sequences: A103445 A103446 A103447 this_sequence A103449 A103450 A103451
%Y A103448 Sequence in context: A058604 A072661 A103432 this_sequence A078803 A130403 A130402
%K A103448 sign
%O A103448 0,2
%A A103448 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 07 2005
%I A078803
%S A078803 1,1,1,1,2,1,0,3,3,1,0,2,6,4,1,0,1,7,10,5,1,0,0,6,16,15,6,1,0,0,3,19,
%T A078803 30,21,7,1,0,0,1,16,45,50,28,8,1,0,0,0,10,51,90,77,36,9,1,0,0,0,4,45,
%U A078803 126,161,112,45,10,1,0,0,0,1,30,141,266,266,156,55,11,1,0,0,0,0,15,126
%N A078803 Triangular array T given by T(n,k)= number of compositions of n into k parts, each in the set {1,2,3}.
%C A078803 Reversing the rows produces A078802. Row sums: A000073.
%C A078803 Number of tribonacci binary words of length n-1 having k-1 1's. A tribonacci binary word is a binary word having no three consecutive 0's. Example: T(6,3)=7 because we have 00101,00110,01001,01010,01100,10010,and 10100. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 16 2007
%D A078803 C. Kimberling, Binary Words with Restricted Repetitions and Associated Compositions of Integers, preprint.
%F A078803 T(n, k)=t(n-1, n-k), for 1<=k<=n, for n>=1, where array t is given by A078802.
%F A078803 G.f.: 1/[1-tz(1+z+z^2)]-1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 10 2004
%e A078803 T(5,2)=2 counts the compositions 2+3 and 3+2. Top of triangle T:
%e A078803 1
%e A078803 1 1
%e A078803 1 2 1
%e A078803 0 3 3 1
%e A078803 0 2 6 4 1
%Y A078803 Cf. A027907, A078802.
%Y A078803 Adjacent sequences: A078800 A078801 A078802 this_sequence A078804 A078805 A078806
%Y A078803 Sequence in context: A072661 A103432 A103448 this_sequence A130403 A130402 A089840
%K A078803 nonn,tabl
%O A078803 1,5
%A A078803 Clark Kimberling (ck6(AT)evansville.edu), Dec 06 2002
%E A078803 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 16 2007
%I A130403
%S A130403 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,7,3,2,1,0,6,8,4,3,2,1,0,7,6,5,4,3,2,1,
%T A130403 0,8,4,7,5,4,3,2,1,0,9,5,6,6,5,4,3,2,1,0,10,17,8,8,8,5,4,3,2,1,0,11,18,
%U A130403 9,7,6,8,5,5,3,2,1,0,12,20,10,9,7,7,7,4,4,3,2,1,0,13,21,12,10,9,6
%N A130403 Signature permutations of SPINE-transformations of A057163-conjugates of Catalan automorphisms in table A122204.
%C A130403 Row n is the signature permutation of the Catalan automorphism which is obtained from A057163-conjugate of the n-th automorphism in the table A122204 with the recursion scheme "SPINE", i.e. row n is obtained as SPINE(A057163 o ENIPS(A089840[n]) o A057163). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A130402. This table contains also all the rows of A122203 and A089840.
%Y A130403 Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A082345, 2: A130936, 3: A073288, 4: A130942, 5: A130940, 6: A130938, 7: A130944, 8: A130946, 9: A130952, 10: A130950, 11: A130948, 12: A057161, 13: A130962, 14: A130964, 15: A069767, 16: A130966, 17: A074688, 18: A130954, 19: A130956, 20: A130960, 21: A130958, Other rows: 169: A069770, 3617: A082339, 65167: A057501.
%Y A130403 Cf. also tables A089840, A122201-A122204, A122285-A122286, A130400-A130401.
%Y A130403 Cf. As a sequence differs from A130403 for the first time at n=92, where a(n)=21, while A130403(n)=22.
%Y A130403 Adjacent sequences: A130400 A130401 A130402 this_sequence A130404 A130405 A130406
%Y A130403 Sequence in context: A103432 A103448 A078803 this_sequence A130402 A089840 A130400
%K A130403 nonn,tabl
%O A130403 0,4
%A A130403 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jun 11 2007
%I A130402
%S A130402 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,7,3,2,1,0,6,8,4,3,2,1,0,7,6,5,4,3,2,1,
%T A130402 0,8,4,7,5,4,3,2,1,0,9,5,6,6,5,4,3,2,1,0,10,17,8,8,8,5,4,3,2,1,0,11,18,
%U A130402 9,7,6,8,5,5,3,2,1,0,12,20,10,9,7,7,7,4,4,3,2,1,0,13,22,12,10,9,6
%N A130402 Signature permutations of ENIPS-transformations of A057163-conjugates of Catalan automorphisms in table A122203.
%C A130402 Row n is the signature permutation of the Catalan automorphism which is obtained from A057163-conjugate of the n-th automorphism in the table A122203 with the recursion scheme "ENIPS", i.e. row n is obtained as ENIPS(A057163 o SPINE(A089840[n]) o A057163). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A130403. This table contains also all the rows of A122204 and A089840.
%Y A130402 Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A082346, 2: A130935, 3: A073289, 4: A130937, 5: A130939, 6: A130941, 7: A130943, 8: A130945, 9: A130947, 10: A130949, 11: A130951, 12: A074687, 13: A130953, 14: A130955, 15: A130957, 16: A130959, 17: A057162, 18: A130961, 19: A130963, 20: A130965, 21: A069768. Other rows: 251: A069770, 3613: A082340, 65352: A057502.
%Y A130402 Cf. also tables A089840, A122201-A122204, A122285-A122286, A130400-A130401.
%Y A130402 Cf. As a sequence differs from A130403 for the first time at n=92, where a(n)=22, while A130403(n)=21.
%Y A130402 Adjacent sequences: A130399 A130400 A130401 this_sequence A130403 A130404 A130405
%Y A130402 Sequence in context: A103448 A078803 A130403 this_sequence A089840 A130400 A130401
%K A130402 nonn,tabl
%O A130402 0,4
%A A130402 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jun 11 2007
%I A089840
%S A089840 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,7,3,2,1,0,6,8,4,3,2,1,0,7,6,6,5,3,2,1,
%T A089840 0,8,4,5,4,5,3,2,1,0,9,5,7,6,6,6,3,2,1,0,10,17,8,7,4,5,6,3,2,1,0,11,18,
%U A089840 9,8,7,4,4,4,3,2,1,0,12,20,10,12,8,7,5,5,4,3,2,1,0,13,21,14,13,12,8,7,6
%N A089840 Signature permutations of non-recursive Catalan automorphisms, sorted according to the minimum number of opening nodes needed in their defining clauses.
%C A089840 Each row is a permutation of natural numbers and occurs only once. The table is closed with regards to the composition of its rows (see A089839) and it contains the inverse of each (their positions are shown in A089843). The permutations in table form an enumerable subgroup of the group of all "Catalan automorphisms" (automorphisms of finite unlabeled rooted plane binary trees). The order of each element is shown at A089842.
%D A089840 A. Karttunen, paper in preparation, draft available by e-mail.
%H A089840 A. Karttunen, C-program for computing the terms of this table. Defines also the order of rows.
%H A089840 A. Karttunen, Prolog-program which illustrates the construction of each row
%Y A089840 The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069770, 2: A072796, 3: A089850, 4: A089851, 5: A089852, 6: A089853, 7: A089854, 8: A072797, 9: A089855, 10: A089856, 11: A089857, 12: A074679, 13: A089858, 14: A073269, 15: A089859, 16: A089860, 17: A074680, 18: A089861, 19: A073270, 20: A089862, 21: A089863.
%Y A089840 Other rows: row 253: A123503, row 258: A123499, row 264: A123500, row 3702: A082354, row 3886: A082353, row 4069: A082351, row 4207: A089865, row 4253: A082352, row 4299: A089866, row 65518: A123495, row 65796: A123496, row 79361: A123492, row 1653002: A123695, row 1653063: A123696, row 1654023: A073281, row 1654249: A123498, row 1654694: A089864, row 1655089: A123497, row 1783367: A123713, row 1786785: A123714.
%Y A089840 Tables A122200, A122201, A122202, A122203, A122204, A122283, A122284, A122285, A122286, A122287, A122288, A122289, A122290 give various "recursive derivations" of these non-recursive automorphisms. See also A089831, A073200.
%Y A089840 Adjacent sequences: A089837 A089838 A089839 this_sequence A089841 A089842 A089843
%Y A089840 Sequence in context: A078803 A130403 A130402 this_sequence A130400 A130401 A122289
%K A089840 nonn,tabl
%O A089840 0,4
%A A089840 Antti Karttunen (His_Firstname.His_Surname(AT)gmail.com), Dec 05 2003
%I A130400
%S A130400 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,7,3,2,1,0,6,8,4,3,2,1,0,7,6,6,5,3,2,1,
%T A130400 0,8,4,5,4,5,3,2,1,0,9,5,7,6,6,6,3,2,1,0,10,17,8,7,4,5,6,3,2,1,0,11,18,
%U A130400 9,8,7,4,4,4,3,2,1,0,12,20,11,12,8,7,5,5,4,3,2,1,0,13,21,14,13,12
%N A130400 Signature permutations of INORDER-transformations of non-recursive Catalan automorphisms in table A089840.
%C A130400 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "INORDER". In this recursion scheme the given automorphism is applied at the root of binary tree after the algorithm has recursed down the car-branch (the left hand side tree in the context of binary trees), but before the algorithm recurses down to the cdr-branch (the right hand side of the binary tree, with respect to the new orientation of branches, possibly changed by the applied automorphism). I.e. this corresponds to the depth-first in-order traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures INORDER and !INORDER can be used to obtain such a transformed automorphism from any constructively (or respectively: destructively) implemented automorphism. Each row occurs only once in this table, and similar notes as given e.g. for table A122202 apply here, e.g. the rows of A089840 all occur here as well. This transformation has many fixed points besides the trivial identity automorphism *A001477: at least *A069770, *A089863 and *A129604 stay as they are. Inverses of these permutations can be found in table A130401.
%o A130400 (Scheme:) (define (INORDER f) (letrec ((g (lambda (s) (cond ((not (pair? s)) s) (else (let ((t (f (cons (g (car s)) (cdr s))))) (cons (car t) (g (cdr t))))))))) g))
%o A130400 (define (!INORDER f!) (letrec ((g! (lambda (s) (cond ((pair? s) (g! (car s)) (f! s) (g! (cdr s)))) s))) g!))
%Y A130400 Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069770, 2: A073284, 3: A122341, 4: A130381, 5: A130383, 6: A130385, 7: A122350, 8: A082341, 9: A130387, 10: A130389, 11: A130391, 13: A130393, 14: A130395, 15: A130397, 16: A130927, 17: A071657, 18: A130929, 19: A130931, 20: A130933, 21: A089863. Other rows: row 1654694: A073280, row 1654720: A129604.
%Y A130400 Cf. also tables A089840, A122201-A122204, A130402-A130403.
%Y A130400 Cf. As a sequence differs from A130401 for the first time at n=80, where a(n)=11, while A130401(n)=14.
%Y A130400 Adjacent sequences: A130397 A130398 A130399 this_sequence A130401 A130402 A130403
%Y A130400 Sequence in context: A130403 A130402 A089840 this_sequence A130401 A122289 A122290
%K A130400 nonn,tabl
%O A130400 0,4
%A A130400 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jun 11 2007
%I A130401
%S A130401 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,7,3,2,1,0,6,8,4,3,2,1,0,7,6,6,5,3,2,1,
%T A130401 0,8,4,5,4,5,3,2,1,0,9,5,7,6,6,6,3,2,1,0,10,17,8,7,4,5,6,3,2,1,0,11,18,
%U A130401 9,8,7,4,4,4,3,2,1,0,12,20,14,13,8,7,5,5,4,3,2,1,0,13,21,10,12,13
%N A130401 Signature permutations of REDRONI-transformations of non-recursive Catalan automorphisms in table A089840.
%C A130401 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "REDRONI". In this recursion scheme the given automorphism is applied at the root of binary tree after the algorithm has recursed down the cdr-branch (the right hand side tree in the context of binary trees), but before the algorithm recurses down to the car-branch (the left hand side of the binary tree, with respect to the new orientation of branches, possibly changed by the applied automorphism). I.e. this corresponds to the reversed depth-first in-order traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures REDRONI and !REDRONI can be used to obtain such a transformed automorphism from any constructively (or respectively: destructively) implemented automorphism. Each row occurs only once in this table, and similar notes as given e.g. for table A122202 apply here, e.g. the rows of A089840 all occur here as well. This transformation has many fixed points besides the trivial identity automorphism *A001477: at least *A069770, *A089859 and *A129604 stay as they are. Inverses of these permutations can be found in table A130400.
%o A130401 (MIT Scheme:) (define (REDRONI f) (letrec ((g (lambda (s) (fold-right (lambda (x y) (let ((t (f (cons x y)))) (cons (g (car t)) (cdr t)))) '() s)))) g))
%o A130401 (define (!REDRONI f!) (letrec ((g! (lambda (s) (cond ((pair? s) (g! (cdr s)) (f! s) (g! (car s)))) s))) g!))
%Y A130401 Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069770, 2: A073285, 3: A122342, 4: A130386, 5: A130384, 6: A130382, 7: A122349, 8: A082342, 9: A130392, 10: A130390, 11: A130388, 12: A071658, 13: A130930, 14: A130932, 15: A089859, 16: A130934, 18: A130394, 19: A130396, 20: A130928, 21: A130398. Other rows: row 1654694: A073280, row 1654720: A129604.
%Y A130401 Cf. also tables A089840, A122201-A122204, A130402-A130403.
%Y A130401 Cf. As a sequence differs from A130400 for the first time at n=80, where a(n)=14, while A130401(n)=11.
%Y A130401 Adjacent sequences: A130398 A130399 A130400 this_sequence A130402 A130403 A130404
%Y A130401 Sequence in context: A130402 A089840 A130400 this_sequence A122289 A122290 A122284
%K A130401 nonn,tabl
%O A130401 0,4
%A A130401 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Jun 11 2007
%I A122289
%S A122289 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,7,3,2,1,0,6,8,4,3,2,1,0,7,6,6,5,3,2,
%T A122289 1,0,8,4,5,4,5,3,2,1,0,9,5,7,6,6,6,3,2,1,0,10,18,8,7,4,5,6,3,2,1,0,11,
%U A122289 17,9,8,7,4,4,4,3,2,1,0,12,20,10,12,8,7,5,5,4,3,2,1,0,13,22,14,13,12
%N A122289 Signature permutations of FORK-transformations of Catalan automorphisms in table A122201.
%C A122289 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122201 with the recursion scheme "FORK", or equivalently row n is obtained as FORK(FORK(nth row of A089840)). See A122201 for the description of FORK. Each row occurs only once in this table. Inverses of these permutations can be found in table A122290.
%D A122289 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122289 Index entries for signature-permutations of Catalan automorphisms
%Y A122289 The known rows of this table: row 0 (identity permutation): A001477, row 1: A122351, row 2: A122363. See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122285-A122288.
%Y A122289 Adjacent sequences: A122286 A122287 A122288 this_sequence A122290 A122291 A122292
%Y A122289 Sequence in context: A089840 A130400 A130401 this_sequence A122290 A122284 A122203
%K A122289 nonn,tabl
%O A122289 0,4
%A A122289 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006
%I A122290
%S A122290 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,7,3,2,1,0,6,8,4,3,2,1,0,7,6,6,5,3,2,
%T A122290 1,0,8,4,5,4,5,3,2,1,0,9,5,7,6,6,6,3,2,1,0,10,18,8,7,4,5,6,3,2,1,0,11,
%U A122290 17,9,8,7,4,4,4,3,2,1,0,12,20,10,12,8,7,5,5,4,3,2,1,0,13,22,14,13,15
%N A122290 Signature permutations of KROF-transformations of Catalan automorphisms in table A122202.
%C A122290 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122202 with the recursion scheme "KROF", or equivalently row n is obtained as KROF(KROF(nth row of A089840)). See A122202 for the description of KROF. Each row occurs only once in this table. Inverses of these permutations can be found in table A122289.
%D A122290 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122290 Index entries for signature-permutations of Catalan automorphisms
%Y A122290 The known rows of this table: row 0 (identity permutation): A001477, row 1: A122351, row 2: A122364. See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122285-A122288.
%Y A122290 Adjacent sequences: A122287 A122288 A122289 this_sequence A122291 A122292 A122293
%Y A122290 Sequence in context: A130400 A130401 A122289 this_sequence A122284 A122203 A122287
%K A122290 nonn,tabl
%O A122290 0,4
%A A122290 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006
%I A122284
%S A122284 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,7,3,2,1,0,6,8,4,3,2,1,0,7,6,6,5,3,2,
%T A122284 1,0,8,5,5,4,5,3,2,1,0,9,4,7,6,6,6,3,2,1,0,10,17,8,7,4,5,6,3,2,1,0,11,
%U A122284 18,9,8,7,4,4,4,3,2,1,0,12,20,10,12,8,7,5,5,4,3,2,1,0,13,22,14,13,12
%N A122284 Signature permutations of NEPEED-transformations of non-recursive Catalan automorphisms in table A089840.
%C A122284 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "NEPEED". In this recursion scheme the algorithm first recurses down to all subtrees, before the given automorphism is applied at the root of general tree. I.e. this corresponds to the post-order (postfix) traversal of a Catalan structure, when it is interpreted as a general tree. The associated Scheme-procedures NEPEED and !NEPEED can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism. Each row occurs only once in this table. Inverses of these permutations can be found in table A122283.
%C A122284 The recursion scheme KROF (described in A122202) is equivalent to a composition of recursion schemes ENIPS (described in A122204) and NEPEED, i.e. KROF(f) = NEPEED(ENIPS(f)) holds for all Catalan automorphisms f. Because of the "universal property of folds", these recursion schemes have well-defined inverses, that is, they are bijective mappings on the set of all Catalan automorphisms. Thus we can equivalently define that NEPEED(f) = KROF(ENIPS^{-1}(f)). Specifically, if g = ENIPS(f), then (f s) = (g (cons (car s) (g^{-1} (cdr s)))), that is, to obtain an automorphism f which gives g when subjected to recursion scheme ENIPS, we compose g with its own inverse applied to the cdr-branch of a S-expression (i.e. the right subtree in the context of binary trees). This implies that for any non-recursive automorphism f in the table A089840, ENIPS^{-1}(f) is also in A089840, which in turn implies that the rows of table A122284 form a (proper) subset of the rows of table A122202. E.g. row 1 of A122284 is row 15 of A122202, row 2 of A122284 is row 3617 of A122202, row 12 of A122284 is row 65167 of A122202, row 15 of A122284 is row 169 of A122202. - Antti Karttunen, 25 May 2007
%C A122284 The recursion scheme FORK (described in A122201) is equivalent to a composition of recursion schemes SPINE (described in A122203) and DEEPEN, i.e. FORK(f) = DEEPEN(SPINE(f)) holds for all Catalan automorphisms f. These recursion schemes have well-defined inverses, that is, they are bijective mappings on the set of all Catalan automorphisms. Thus we can equivalently define that DEEPEN(f) = FORK(SPINE^{-1}(f)). Specifically, if g = SPINE(f), then (f s) = (cond ((pair? s) (let ((t (g s))) (cons (car t) (g^{-1} (cdr t))))) (else s)) that is, to obtain an automorphism f which gives g when subjected to recursion scheme SPINE, we compose g with its own inverse applied to the cdr-branch of a S-expression. This implies that for any non-recursive automorphism f in the table A089840, SPINE^{-1}(f) is also in A089840, which in turn implies that the rows of table A122283 form a (proper) subset of the rows of table A122201. E.g. row 1 of A122283 is row 21 of A122201, row 2 of A122283 is row 3613 of A122201, row 17 of A122283 is row 65352 of A122201, row 21 of A122283 is row 251 of A122201. - Antti Karttunen, 25 May 2007
%D A122284 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122284 Index entries for signature-permutations of Catalan automorphisms
%o A122284 (Scheme:) (define (NEPEED foo) (letrec ((bar (lambda (s) (foo (map bar s))))) bar))
%o A122284 (define (!NEPEED foo!) (letrec ((bar! (lambda (s) (for-each bar! s) (foo! s) s))) bar!))
%Y A122284 The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A122302, 2: A122300, 3: A122304, 4: A122310, 5: A122308, 6: A122306, 7: A122312, 8: A122314, 9: A122320, 10: A122318, 11: A122316, 12: A122332, 13: A122334, 14: A122336, 15: A122340, 16: A122338, 17: A122322, 18: A122324, 19: A122326, 20: A122330, 21: A122328. See also tables A089840, A122200, A122201-A122204, A122285-A122288, A122289-A122290.
%Y A122284 Adjacent sequences: A122281 A122282 A122283 this_sequence A122285 A122286 A122287
%Y A122284 Sequence in context: A130401 A122289 A122290 this_sequence A122203 A122287 A122283
%K A122284 nonn,tabl
%O A122284 0,4
%A A122284 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006
%I A122203
%S A122203 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,7,3,2,1,0,6,8,4,3,2,1,0,7,6,6,5,3,2,1,
%T A122203 0,8,5,5,4,5,3,2,1,0,9,4,7,6,6,6,3,2,1,0,10,17,8,7,4,5,6,3,2,1,0,11,18,
%U A122203 9,8,7,4,4,4,3,2,1,0,12,20,11,12,8,7,5,5,4,3,2,1,0,13,21,14,13,12
%N A122203 Signature permutations of SPINE-transformations of non-recursive Catalan automorphisms in table A089840.
%C A122203 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "SPINE". In this recursion scheme the given automorphism is first applied at the root of binary tree, before the algorithm recurses down to the new right-hand side branch. The associated Scheme-procedures SPINE and !SPINE can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism. Each row occurs only once in this table. Inverses of these permutations can be found in table A122204.
%C A122203 The recursion scheme SPINE has a well-defined inverse, that is, it acts as a bijective mapping on the set of all Catalan automorphisms. Specifically, if g = SPINE(f), then (f s) = (cond ((pair? s) (let ((t (g s))) (cons (car t) (g^{-1} (cdr t))))) (else s)) that is, to obtain an automorphism f which gives g when subjected to recursion scheme SPINE, we compose g with its own inverse applied to the cdr-branch of a S-expression. This implies that for any non-recursive automorphism f in the table A089840, SPINE^{-1}(f) is also in A089840, which in turn implies that the rows of table A089840 form a (proper) subset of the rows of this table.
%D A122203 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122203 Index entries for signature-permutations of Catalan automorphisms
%o A122203 (Scheme:) (define (SPINE foo) (letrec ((bar (lambda (s) (let ((t (foo s))) (if (pair? t) (cons (car t) (bar (cdr t))) t))))) bar))
%o A122203 (define (!SPINE foo!) (letrec ((bar! (lambda (s) (cond ((pair? s) (foo! s) (bar! (cdr s)))) s))) bar!))
%Y A122203 Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069767, 2: A057509, 3: A130341, 4: A130343, 5: A130345, 6: A130347, 7: A122282, 8: A082339, 9: A130349, 10: A130351, 11: A130353, 12: A074685, 13: A130355, 14: A130357, 15: A130359, 16: A130361, 17: A057501, 18: A130363, 19: A130365, 20: A130367, 21: A069770. Other rows: row 251: A089863, row 253: A123717, row 3608: A129608, row 3613: A072796, row 65352: A074680, row 79361: A123715.
%Y A122203 See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122285-A122288, A122289-A122290.
%Y A122203 Adjacent sequences: A122200 A122201 A122202 this_sequence A122204 A122205 A122206
%Y A122203 Sequence in context: A122289 A122290 A122284 this_sequence A122287 A122283 A122204
%K A122203 nonn,tabl
%O A122203 0,4
%A A122203 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006, Jun 06 2007
%I A122287
%S A122287 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,7,3,2,1,0,6,8,4,3,2,1,0,7,6,6,5,3,2,1,
%T A122287 0,8,5,5,4,5,3,2,1,0,9,4,7,6,6,6,3,2,1,0,10,17,8,7,4,5,6,3,2,1,0,11,18,
%U A122287 9,8,7,4,4,4,3,2,1,0,12,20,14,13,8,7,5,5,4,3,2,1,0,13,21,11,12,13
%N A122287 Signature permutations of FORK-transformations of Catalan automorphisms in table A122204.
%C A122287 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122204 with the recursion scheme "FORK", or equivalently row n is obtained as FORK(ENIPS(nth row of A089840)). See A122201 and A122204 for the description of FORK and ENIPS. Moreover, each row of A122287 can be obtained as the "DEEPEN" transform of the corresponding row in A122286. (See A122283 for the description of DEEPEN). Each row occurs only once in this table. Inverses of these permutations can be found in table A122288. This table contains also all the rows of A122201 and A089840.
%D A122287 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122287 Index entries for signature-permutations of Catalan automorphisms
%Y A122287 The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069767, 2: A057164, 3: A130981, 4: A130983, 5: A130982, 6: A130984, 7: A130986, 8: A130988, 9: A130994, 10: A130992, 11: A130990, 12: A057506, 13: A131004, 14: A131006, 15: A057163, 16: A131008, 17: A131010, 18: A130996, 19: A130998, 20: A131002, 21: A131000. Other rows: 169: A122353, 3617: A057511, 65167: A074681.
%Y A122287 See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122285-A122288, A122289-A122290, A130400-A130403.
%Y A122287 Adjacent sequences: A122284 A122285 A122286 this_sequence A122288 A122289 A122290
%Y A122287 Sequence in context: A122290 A122284 A122203 this_sequence A122283 A122204 A122288
%K A122287 nonn,tabl
%O A122287 0,4
%A A122287 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006, Jun 20 2007
%I A122283
%S A122283 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,8,3,2,1,0,6,7,4,3,2,1,0,7,6,6,5,3,2,
%T A122283 1,0,8,4,5,4,5,3,2,1,0,9,5,7,6,6,6,3,2,1,0,10,22,8,7,4,5,6,3,2,1,0,11,
%U A122283 21,9,8,7,4,4,4,3,2,1,0,12,20,10,12,8,7,5,5,4,3,2,1,0,13,17,14,13,12
%N A122283 Signature permutations of DEEPEN-transformations of non-recursive Catalan automorphisms in table A089840.
%C A122283 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "DEEPEN". In this recursion scheme the given automorphism is first applied at the root of general tree, before the algorithm recurses down to all subtrees. I.e. this corresponds to the pre-order (prefix) traversal of a Catalan structure, when it is interpreted as a general tree. The associated Scheme-procedures DEEPEN and !DEEPEN can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism. Each row occurs only once in this table. Inverses of these permutations can be found in table A122284.
%C A122283 The recursion scheme FORK (described in A122201) is equivalent to a composition of recursion schemes SPINE (described in A122203) and DEEPEN, i.e. FORK(f) = DEEPEN(SPINE(f)) holds for all Catalan automorphisms f. These recursion schemes have well-defined inverses, that is, they are bijective mappings on the set of all Catalan automorphisms. Thus we can equivalently define that DEEPEN(f) = FORK(SPINE^{-1}(f)). Specifically, if g = SPINE(f), then (f s) = (cond ((pair? s) (let ((t (g s))) (cons (car t) (g^{-1} (cdr t))))) (else s)) that is, to obtain an automorphism f which gives g when subjected to recursion scheme SPINE, we compose g with its own inverse applied to the cdr-branch of a S-expression. This implies that for any non-recursive automorphism f in the table A089840, SPINE^{-1}(f) is also in A089840, which in turn implies that the rows of table A122283 form a (proper) subset of the rows of table A122201. E.g. row 1 of A122283 is row 21 of A122201, row 2 of A122283 is row 3613 of A122201, row 17 of A122283 is row 65352 of A122201, row 21 of A122283 is row 251 of A122201. - Antti Karttunen, 25 May 2007
%D A122283 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122283 Index entries for signature-permutations of Catalan automorphisms
%o A122283 (Scheme:) (define (DEEPEN foo) (letrec ((bar (lambda (s) (map bar (foo s))))) bar))
%o A122283 (define (!DEEPEN foo!) (letrec ((bar! (lambda (s) (foo! s) (for-each bar! s) s))) bar!))
%Y A122283 The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A122301, 2: A122300, 3: A122303, 4: A122305, 5: A122307, 6: A122309, 7: A122311, 8: A122313, 9: A122315, 10: A122317, 11: A122319, 12: A122321, 13: A122323, 14: A122325, 15: A122327, 16: A122329, 17: A122331, 18: A122333, 19: A122335, 20: A122337, 21: A122339. See also tables A089840, A122200, A122201-A122204, A122285-A122288, A122289-A122290.
%Y A122283 Adjacent sequences: A122280 A122281 A122282 this_sequence A122284 A122285 A122286
%Y A122283 Sequence in context: A122284 A122203 A122287 this_sequence A122204 A122288 A122201
%K A122283 nonn,tabl
%O A122283 0,4
%A A122283 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006
%I A122204
%S A122204 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,8,3,2,1,0,6,7,4,3,2,1,0,7,6,6,5,3,2,1,
%T A122204 0,8,4,5,4,5,3,2,1,0,9,5,7,6,6,6,3,2,1,0,10,22,8,7,4,5,6,3,2,1,0,11,21,
%U A122204 9,8,7,4,4,4,3,2,1,0,12,20,14,13,8,7,5,5,4,3,2,1,0,13,17,10,12,13
%N A122204 Signature permutations of ENIPS-transformations of non-recursive Catalan automorphisms in table A089840.
%C A122204 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "ENIPS". In this recursion scheme the algorithm first recurses down to the right-hand side branch of the binary tree, before the given automorphism is applied at its root. This corresponds to the fold-right operation applied to the Catalan structure, interpreted e.g. as a parenthesization or a Lisp-like list, where (lambda (x y) (f (cons x y))) is the binary function given to fold, with 'f' being the given automorphism. The associated Scheme-procedures ENIPS and !ENIPS can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism. Each row occurs only once in this table. Inverses of these permutations can be found in table A122203.
%C A122204 Because of the "universal property of folds", the recursion scheme ENIPS has a well-defined inverse, that is, it acts as a bijective mapping on the set of all Catalan automorphisms. Specifically, if g = ENIPS(f), then (f s) = (g (cons (car s) (g^{-1} (cdr s)))), that is, to obtain an automorphism f which gives g when subjected to recursion scheme ENIPS, we compose g with its own inverse applied to the cdr-branch of a S-expression (i.e. the right subtree in the context of binary trees). This implies that for any non-recursive automorphism f in the table A089840, ENIPS^{-1}(f) is also in A089840, which in turn implies that the rows of table A089840 form a (proper) subset of the rows of this table.
%D A122204 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122204 Index entries for signature-permutations of Catalan automorphisms
%o A122204 (MIT Scheme:) (define (ENIPS foo) (lambda (s) (fold-right (lambda (x y) (foo (cons x y))) '() s)))
%o A122204 (define (!ENIPS foo!) (letrec ((bar! (lambda (s) (cond ((pair? s) (bar! (cdr s)) (foo! s))) s))) bar!))
%Y A122204 Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069768, 2: A057510, 3: A130342, 4: A130348, 5: A130346, 6: A130344, 7: A122282, 8: A082340, 9: A130354, 10: A130352, 11: A130350, 12: A057502, 13: A130364, 14: A130366, 15: A069770, 16: A130368, 17: A074686, 18: A130356, 19: A130358, 20: A130362, 21: A130360. Other rows: row 169: A089859, row 253: A123718, row 3608: A129608, row 3613: A072796, row 65167: A074679, row 79361: A123716.
%Y A122204 See also tables A089840, A122200, A122201-A122203, A122283-A122284, A122285-A122288, A122289-A122290.
%Y A122204 Adjacent sequences: A122201 A122202 A122203 this_sequence A122205 A122206 A122207
%Y A122204 Sequence in context: A122203 A122287 A122283 this_sequence A122288 A122201 A122286
%K A122204 nonn,tabl
%O A122204 0,4
%A A122204 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006, Jun 06 2007
%I A122288
%S A122288 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,8,3,2,1,0,6,7,4,3,2,1,0,7,6,6,5,3,2,1,
%T A122288 0,8,4,5,4,5,3,2,1,0,9,5,7,6,6,6,3,2,1,0,10,22,8,7,4,5,6,3,2,1,0,11,21,
%U A122288 9,8,7,4,4,4,3,2,1,0,12,20,14,13,8,7,5,5,4,3,2,1,0,13,17,11,12,13
%N A122288 Signature permutations of KROF-transformations of Catalan automorphisms in table A122203.
%C A122288 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122203 with the recursion scheme "KROF", or equivalently row n is obtained as KROF(SPINE(nth row of A089840)). See A122202 and A122203 for the description of KROF and SPINE. Moreover, each row of A122288 can be obtained as the "NEPEED" transform of the corresponding row in A122285. (See A122284 for the description of NEPEED). Each row occurs only once in this table. Inverses of these permutations can be found in table A122287. This table contains also all the rows of A122202 and A089840.
%D A122288 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122288 Index entries for signature-permutations of Catalan automorphisms
%Y A122288 The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069768, 2: A057164, 3: A130981, 4: A130983, 5: A130982, 6: A130984, 7: A130985, 8: A130987, 9: A130989, 10: A130991, 11: A130993, 12: A131009, 13: A130995, 14: A130997, 15: A130999, 16: A131001, 17: A057505, 18: A131003, 19: A131005, 20: A131007, 21: A057163. Other rows: 251: A122354, 3613: A057512, 65352: A074682.
%Y A122288 See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122285-A122286, A122289-A122290, A130400-A130403.
%Y A122288 Adjacent sequences: A122285 A122286 A122287 this_sequence A122289 A122290 A122291
%Y A122288 Sequence in context: A122287 A122283 A122204 this_sequence A122201 A122286 A122202
%K A122288 nonn,tabl
%O A122288 0,4
%A A122288 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006, Jun 20 2007
%I A122201
%S A122201 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,8,3,2,1,0,6,7,4,3,2,1,0,7,6,6,5,3,2,
%T A122201 1,0,8,5,5,4,5,3,2,1,0,9,4,7,6,6,6,3,2,1,0,10,22,8,7,4,5,6,3,2,1,0,11,
%U A122201 21,9,8,7,4,4,4,3,2,1,0,12,20,11,12,8,7,5,5,4,3,2,1,0,13,18,14,13,12
%N A122201 Signature permutations of FORK-transformations of non-recursive Catalan automorphisms in table A089840.
%C A122201 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "FORK". In this recursion scheme the given automorphism is first applied at the root of binary tree, before the algorithm recurses down to the both branches (new ones, possibly changed by the given automorphism). I.e. this corresponds to the pre-order (prefix) traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures FORK and !FORK can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism. Each row occurs only once in this table. Inverses of these permutations can be found in table A122202.
%D A122201 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122201 Index entries for signature-permutations of Catalan automorphisms
%o A122201 (Scheme:) (define (FORK foo) (letrec ((bar (lambda (s) (let ((t (foo s))) (if (pair? t) (cons (bar (car t)) (bar (cdr t))) t))))) bar))
%o A122201 (define (!FORK foo!) (letrec ((bar! (lambda (s) (cond ((pair? s) (foo! s) (bar! (car s)) (bar! (cdr s)))) s))) bar!))
%Y A122201 The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A057163, 2: A057511, 3: A122341, 4: A122343, 5: A122345, 6: A122347, 7: A122349, 8: A082325, 9: A082360, 10: A122291, 11: A122293, 12: A074681, 13: A122295, 14: A122297, 15: A122353, 16: A122355, 17: A074684, 18: A122357, 19: A122359, 20: A122361, 21: A122301. Other rows: row 4253: A082356, row 65796: A082358, row 79361: A123493.
%Y A122201 See also tables A089840, A122200, A122202-A122204, A122283-A122284, A122285-A122288, A122289-A122290.
%Y A122201 Adjacent sequences: A122198 A122199 A122200 this_sequence A122202 A122203 A122204
%Y A122201 Sequence in context: A122283 A122204 A122288 this_sequence A122286 A122202 A122285
%K A122201 nonn,tabl
%O A122201 0,4
%A A122201 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006
%I A122286
%S A122286 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,8,3,2,1,0,6,7,4,3,2,1,0,7,6,6,5,3,2,1,
%T A122286 0,8,5,5,4,5,3,2,1,0,9,4,7,6,6,6,3,2,1,0,10,22,8,7,4,5,6,3,2,1,0,11,21,
%U A122286 9,8,7,4,4,4,3,2,1,0,12,20,14,13,8,7,5,5,4,3,2,1,0,13,17,11,12,13
%N A122286 Signature permutations of SPINE-transformations of Catalan automorphisms in table A122204.
%C A122286 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122204 with the recursion scheme "SPINE", or equivalently row n is obtained as SPINE(ENIPS(nth row of A089840)). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A122285. This table contains also all the rows of A122203 and A089840.
%D A122286 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122286 Index entries for signature-permutations of Catalan automorphisms
%Y A122286 The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A082347, 2: A057508, 3: A131142, 4: A131148, 5: A131146, 6: A131144, 7: A131173, 8: A131170, 9: A131154, 10: A131152, 11: A131150, 12: A057504, 13: A131164, 14: A131166, 15: A069767, 16: A131168, 17: A131172, 18: A131156, 19: A131158, 20: A131162, 21: A131160. Other rows: row 169: A130359, 3608: A130339, 3617: A057509, 65167: A074685.
%Y A122286 See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122285-A122288, A122289-A122290, A130400-A130403. As a sequence differs from A122285 for the first time at n=92, where a(n)=17, while A122285(n)=18.
%Y A122286 Adjacent sequences: A122283 A122284 A122285 this_sequence A122287 A122288 A122289
%Y A122286 Sequence in context: A122204 A122288 A122201 this_sequence A122202 A122285 A100224
%K A122286 nonn,tabl
%O A122286 0,4
%A A122286 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006, Jun 20 2007
%I A122202
%S A122202 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,8,3,2,1,0,6,7,4,3,2,1,0,7,6,6,5,3,2,
%T A122202 1,0,8,5,5,4,5,3,2,1,0,9,4,7,6,6,6,3,2,1,0,10,22,8,7,4,5,6,3,2,1,0,11,
%U A122202 21,9,8,7,4,4,4,3,2,1,0,12,20,14,13,8,7,5,5,4,3,2,1,0,13,18,10,12,13
%N A122202 Signature permutations of KROF-transformations of non-recursive Catalan automorphisms in table A089840.
%C A122202 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "KROF". In this recursion scheme the algorithm first recurses down to the both branches, before the given automorphism is applied at the root of binary tree. I.e. this corresponds to the post-order (postfix) traversal of a Catalan structure, when it is interpreted as a binary tree. The associated Scheme-procedures KROF and !KROF can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism. Each row occurs only once in this table. Inverses of these permutations can be found in table A122201.
%C A122202 The recursion scheme KROF is equivalent to a composition of recursion schemes ENIPS (described in A122204) and NEPEED (described in A122284), i.e. KROF(f) = NEPEED(ENIPS(f)) holds for all Catalan automorphisms f. Because of the "universal property of folds", these recursion schemes have well-defined inverses, that is, they are bijective mappings on the set of all Catalan automorphisms. Specifically, if g = KROF(f), then (f s) = (g (cons (g^{-1} (car s)) (g^{-1} (cdr s)))), that is, to obtain an automorphism f which gives g when subjected to recursion scheme KROF, we compose g with its own inverse applied to the car- and cdr-branches of a S-expression (i.e. the left and right subtrees in the context of binary trees). This implies that for any non-recursive automorphism f of the table A089840, KROF^{-1}(f) is also in A089840, which in turn implies that all rows of table A089840 can be found also in table A122202 (e.g. the row 1 of A089840 (A069770) occurs here as row 1654720) and furthermore, the table A122290 contains the rows of both tables, A122202 and A089840 as its subsets. Similar notes apply to recursion scheme FORK described in A122201. - Antti Karttunen, 25 May 2007
%D A122202 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122202 Index entries for signature-permutations of Catalan automorphisms
%o A122202 (MIT Scheme:) (define (KROF foo) (letrec ((bar (lambda (s) (fold-right (lambda (x y) (foo (cons (bar x) y))) '() s)))) bar))
%o A122202 (define (!KROF foo!) (letrec ((bar! (lambda (s) (cond ((pair? s) (bar! (car s)) (bar! (cdr s)) (foo! s))) s))) bar!))
%Y A122202 The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A057163, 2: A057512, 3: A122342, 4: A122348, 5: A122346, 6: A122344, 7: A122350, 8: A082326, 9: A122294, 10: A122292, 11: A082359, 12: A074683, 13: A122358, 14: A122360, 15: A122302, 16: A122362, 17: A074682, 18: A122296, 19: A122298, 20: A122356, 21: A122354. Other rows: row 4069: A082355, row 65518: A082357, row 79361: A123494.
%Y A122202 See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122285-A122288, A122289-A122290.
%Y A122202 Row 1654720: A069770.
%Y A122202 Adjacent sequences: A122199 A122200 A122201 this_sequence A122203 A122204 A122205
%Y A122202 Sequence in context: A122288 A122201 A122286 this_sequence A122285 A100224 A089000
%K A122202 nonn,tabl
%O A122202 0,4
%A A122202 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006
%I A122285
%S A122285 0,1,0,2,1,0,3,3,1,0,4,2,2,1,0,5,8,3,2,1,0,6,7,4,3,2,1,0,7,6,6,5,3,2,1,
%T A122285 0,8,5,5,4,5,3,2,1,0,9,4,7,6,6,6,3,2,1,0,10,22,8,7,4,5,6,3,2,1,0,11,21,
%U A122285 9,8,7,4,4,4,3,2,1,0,12,20,14,13,8,7,5,5,4,3,2,1,0,13,18,11,12,13
%N A122285 Signature permutations of ENIPS-transformations of Catalan automorphisms in table A122203.
%C A122285 Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122203 with the recursion scheme "ENIPS", or equivalently row n is obtained as ENIPS(SPINE(nth row of A089840)). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A122286. This table contains also all the rows of A122204 and A089840.
%D A122285 A. Karttunen, paper in preparation, draft available by e-mail.
%H A122285 Index entries for signature-permutations of Catalan automorphisms
%Y A122285 The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A082348, 2: A057508, 3: A131141, 4: A131143, 5: A131145, 6: A131147, 7: A131173, 8: A131169, 9: A131149, 10: A131151, 11: A131153, 12: A131171, 13: A131155, 14: A131157, 15: A131159, 16: A131161, 17: A057503, 18: A131163, 19: A131165, 20: A131167, 21: A069768. Other rows: row 251: A130360, 3608: A130339, 3613: A057510, 65352: A074686.
%Y A122285 See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122286-A122288, A122289-A122290, A130400-A130403. As a sequence differs from A122286 for the first time at n=92, where a(n)=18, while A122286(n)=17.
%Y A122285 Adjacent sequences: A122282 A122283 A122284 this_sequence A122286 A122287 A122288
%Y A122285 Sequence in context: A122201 A122286 A122202 this_sequence A100224 A089000 A107238
%K A122285 nonn,tabl
%O A122285 0,4
%A A122285 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Sep 01 2006, Jun 20 2007
%I A100224
%S A100224 1,1,0,1,0,2,1,0,3,3,1,0,4,4,6,1,0,5,5,10,10,1,0,6,6,15,18,17,1,0,7,7,
%T A100224 21,28,35,28,1,0,8,8,28,40,60,64,46,1,0,9,9,36,54,93,117,117,75,1,0,10,
%U A100224 10,45,70,135,190,230,210,122
%N A100224 Triangle, read by rows, of the coefficients of [x^k] in G100224(x)^n such that the row sums are 2^n-1 for n>0, where G100224(x) is the g.f. of A100224.
%C A100224 Diagonals are: T(n,n)=A001610(n-1) for n>0, with T(0,0)=1, T(n+1,n)=A006490(n), T(n+2,n)=A006491(n), T(n+3,n)=A006492(n), T(n+4,n)=A006493(n). The ratio of the generating functions of any two adjacent diagonals gives: (1-x)/(1-x-x^2) = 1+ x^2+ x^3+ 2*x^4+ 3*x^5+ 5*x^6+ 8*x^7+ 13*x^8+...
%F A100224 G.f.: A(x, y)=(1-2*x*y+2*x^2*y^2)/((1-x*y)*(1-x*y-x^2*y^2-x*(1-x*y))).
%e A100224 Rows begin:
%e A100224 [1],
%e A100224 [1,0],
%e A100224 [1,0,2],
%e A100224 [1,0,3,3],
%e A100224 [1,0,4,4,6],
%e A100224 [1,0,5,5,10,10],
%e A100224 [1,0,6,6,15,18,17],
%e A100224 [1,0,7,7,21,28,35,28],
%e A100224 [1,0,8,8,28,40,60,64,46],...
%e A100224 where row sums form 2^n-1 for n>0:
%e A100224 2^1-1 = 1+0 = 1
%e A100224 2^2-1 = 1+0+2 = 3
%e A100224 2^3-1 = 1+0+3+3 = 7
%e A100224 2^4-1 = 1+0+4+4+6 = 15
%e A100224 2^5-1 = 1+0+5+5+10+10 = 31.
%e A100224 The main diagonal forms A001610 = [0,2,3,6,10,17,...],
%e A100224 where Sum_{n>=1} A001610(n-1)/n*x^n = log((1-x)/(1-x-x^2).
%o A100224 (PARI) {T(n,k)=if(n= 1, R(1, j)=1 for j >= 0, R(i, j)=SUM{R(i-2, k)+R(i-1, k): k=0, 1, ..., j} for i >= 1, j >= 1.
%F A055830 Sum_{k, 0<=k<=n}x^k*T(n,k)= A039834(n-2), A000012(n), A000045(n+1), A001333(n), A003688(n), A015448(n), A015449(n), A015451(n), A015453(n), A015454(n), A015455(n), A015456(n), A015457(n) for x= -2,-1,0,1,2,3,4,5,6,7,8,9,10 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 22 2006
%F A055830 Sum_{k, 0<=k<=[n/2]}T(n-k,k)=A011782(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 22 2006
%F A055830 Triangle T(n,k), 0<=k<=n, given by [1, 1, -1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 2006
%F A055830 T(n,0)= Fibonacci(n+1)=A000045(n+1) . Sum_{k, 0<=k<=n}T(n,k)=A001333(n) . T(n,k)=0 if k>n or if k<0, T(0,0)=1, T(1,1)=0, T(n,k)=T(n-1,k-1)+T(n-1,k)+T(n-2,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 2006
%e A055830 1
%e A055830 1,0
%e A055830 2,1,0
%e A055830 3,3,1,0
%e A055830 5,7,4,1,0
%e A055830 8,15,12,5,1,0
%e A055830 13,30,31,18,6,1,0
%e A055830 21,58,73,54,25,7,1,0
%e A055830 34,109,162,145,85,33,8,1,0
%e A055830 55,201,344,361,255,125,42,9,1,0
%Y A055830 Left-hand columns include A000045, A023610.
%Y A055830 Right-hand columns include A055831, A055832, A055833, A055834, A055835, A055836, A055837, A055838, A055839, A055840.
%Y A055830 Row sums: A001333 (numerators of continued fraction convergents to sqrt(2)).
%Y A055830 Cf. A122075 (another version).
%Y A055830 Adjacent sequences: A055827 A055828 A055829 this_sequence A055831 A055832 A055833
%Y A055830 Sequence in context: A100224 A089000 A107238 this_sequence A079123 A121548 A113020
%K A055830 nonn,tabl
%O A055830 0,4
%A A055830 Clark Kimberling (ck6(AT)evansville.edu), May 28 2000
%E A055830 Edited by Ralf Stephan, Jan 12 2005
%I A079123
%S A079123 0,0,0,0,0,0,0,0,0,1,1,2,1,0,3,3,1,1,0,2,4,5,2,2,4,2,1,8,3,7,9,4,4,6,5,
%T A079123 3,8,2,8,3,9,6,4,6,8,7,7,4,12,4,15,10,10,9,13,10,15,7,7,19,15,9,20,12,
%U A079123 11,12,15,19,12,15,20,10,12,13,13,19,16,13,21,17,17,23,19,13,13,15,18
%N A079123 Number of 4's in n# (n primorial) = 4's in A002110(n).
%o A079123 (PARI) See program in Number of 0's in n#
%Y A079123 Cf. A002110.
%Y A079123 Adjacent sequences: A079120 A079121 A079122 this_sequence A079124 A079125 A079126
%Y A079123 Sequence in context: A089000 A107238 A055830 this_sequence A121548 A113020 A127258
%K A079123 easy,nonn
%O A079123 2,12
%A A079123 Cino Hilliard (hillcino368(AT)gmail.com), Feb 03 2003
%I A121548
%S A121548 1,1,1,1,2,1,0,3,3,1,1,2,6,4,1,0,3,7,10,5,1,0,2,9,16,15,6,1,1,2,9,23,30,
%T A121548 21,7,1,0,2,10,28,50,50,28,8,1,0,3,9,34,71,96,77,36,9,1,0,2,12,36,95,
%U A121548 156,168,112,45,10,1,0,0,12,43,115,231,308,274,156,55,11,1,1,2,9,48,140
%N A121548 Triangle read by rows: T(n,k) is the number of compositions of n into k Fibonacci numbers (1<=k<=n; only one 1 is considered as a Fibonacci number).
%C A121548 Sum of terms in row n = A076739(n). T(n,1)=A010056(n) (the characteristic function of the Fibonacci numbers); T(n,2)=A121549(n); T(n,3)=A121550(n); Sum(k*T(n,k), k=1..n)=A121551(n).
%F A121548 G.f.=G(t,z)=1/[1-t*Sum(z^fibonacci(i), i=2..infinity)]-1.
%e A121548 T(5,3)=6 because we have [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], and [3,1,1].
%e A121548 Triangle starts:
%e A121548 1;
%e A121548 1,1;
%e A121548 1,2,1;
%e A121548 0,3,3,1;
%e A121548 1,2,6,4,1;
%e A121548 0,3,7,10,5,1;
%e A121548 0,2,9,16,15,6,1;
%p A121548 with(combinat): G:=1/(1-t*sum(z^fibonacci(i),i=2..40))-1: Gser:=simplify(series(G,z=0,25)): for n from 1 to 23 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 15 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
%Y A121548 Cf. A076739, A010056, A121549, A121550, A121551.
%Y A121548 Adjacent sequences: A121545 A121546 A121547 this_sequence A121549 A121550 A121551
%Y A121548 Sequence in context: A107238 A055830 A079123 this_sequence A113020 A127258 A049242
%K A121548 nonn,tabl
%O A121548 1,5
%A A121548 Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 07 2006
%I A113020
%S A113020 0,0,1,0,2,1,0,3,3,2,0,4,6,8,3,0,5,10,20,15,5,0,6,15,40,45,30,8,0,7,21,
%T A113020 70,105,105,56,13,0,0,8,28,112,210,280,224,104,21,0,9,36,168,378,630,
%U A113020 672,468,189,34,0,10,45,240,630,1260,1680,1560,945,340,55
%V A113020 0,0,1,0,2,-1,0,3,-3,2,0,4,-6,8,-3,0,5,-10,20,-15,5,0,6,-15,40,-45,30,-8,0,7,-21,70,
%W A113020 -105,105,-56,13,0,0,8,-28,112,-210,280,-224,104,-21,0,9,-36,168,-378,630,-672,468,
%X A113020 -189,34,0,10,-45,240,-630,1260,-1680,1560,-945,340,-55
%N A113020 Number triangle whose row sums are the Fibonacci numbers.
%C A113020 Rows sums are A000045. Diagonal sums are A113021. Main diagonal is (-1)^(n+1)F(n).
%F A113020 T(n, k)=sum{j=0..n, C(n, j)C(0, j-k)F(j-2k)}.
%e A113020 Rows begin
%e A113020 0;
%e A113020 0,1;
%e A113020 0,2,-1;
%e A113020 0,3,-3,2;
%e A113020 0,4,-6,8,-3;
%e A113020 0,5,-10,20,-15,5;
%e A113020 0,6,-15,40,-45,30,-8;
%e A113020 0,7,-21,70,-105,105,-56,13;
%Y A113020 Cf. A094435.
%Y A113020 Adjacent sequences: A113017 A113018 A113019 this_sequence A113021 A113022 A113023
%Y A113020 Sequence in context: A055830 A079123 A121548 this_sequence A127258 A049242 A108887
%K A113020 easy,sign,tabl
%O A113020 0,5
%A A113020 Paul Barry (pbarry(AT)wit.ie), Oct 11 2005
%I A127258
%S A127258 1,1,2,1,0,3,3,2,6,3,4,0,6,4,6,40,105,130,60,18,15,10,0,10,5,24,270,
%T A127258 1350,3925,7260,8712,6485,2445,60,330,18,45,20,0,15,6,120,2016,15750,
%U A127258 75810,250950,603435,1084104,1471305,1502550,1128820,589281,182721
%V A127258 1,-1,2,1,0,-3,3,2,-6,3,4,0,-6,4,-6,40,-105,130,-60,-18,15,10,0,-10,5,-24,270,-1350,
%W A127258 3925,-7260,8712,-6485,2445,60,-330,-18,45,20,0,-15,6,120,-2016,15750,-75810,250950,
%X A127258 -603435,1084104,-1471305,1502550,-1128820,589281,-182721
%N A127258 Triangular array read by rows: B(n,k) (n>=1, 0<=k<=n(n-1)/2) is such that SUM B(n,k)*q^(n*(n-1)/2-k) gives the expectation of the number of connected components in a random graph on n labeled vertices where every edge is present with probability q.
%C A127258 Row-reversed version of A125210, see A125210 for further details.
%e A127258 The array starts with
%e A127258 1
%e A127258 -1, 2
%e A127258 1, 0, -3, 3
%e A127258 2, -6, 3, 4, 0, -6, 4
%e A127258 -6, 40, -105, 130, -60, -18, 15, 10, 0, -10, 5
%e A127258 ...
%Y A127258 Cf. A125210 (row-reversed version), A125209 (dual version).
%Y A127258 Adjacent sequences: A127255 A127256 A127257 this_sequence A127259 A127260 A127261
%Y A127258 Sequence in context: A079123 A121548 A113020 this_sequence A049242 A108887 A095884
%K A127258 sign,tabf
%O A127258 1,3
%A A127258 Max Alekseyev (maxal(AT)cs.ucsd.edu), Jan 09 2007
%I A049242
%S A049242 2,1,0,3,3,3,3,3,3,2,1,0,3,3,3,3,3,3,3,3,3,3,1,3,2,1,0,3,3,3,3,3,3,3,3,
%T A049242 3,3,3,3,3,3,3,3,3,3,2,1,0,3,3,3,3,3,3,3,3,3,2,1,3,3,3,3,3,3,3,3,3,3,3,
%U A049242 1,3,2,1,0,3,3,3,3,3,3,3,1,3,3,3,3,3,3,3,3,3,3,2,3,3,3,3,3,3,3,3,3,3,3
%N A049242 Smallest nonnegative value taken on by 3x^2 - ny^2 for an infinite number of integer pairs (x, y).
%Y A049242 Adjacent sequences: A049239 A049240 A049241 this_sequence A049243 A049244 A049245
%Y A049242 Sequence in context: A121548 A113020 A127258 this_sequence A108887 A095884 A128908
%K A049242 nonn
%O A049242 1,1
%A A049242 David W. Wilson (davidwwilson(AT)comcast.net)
%I A108887
%S A108887 0,1,0,2,1,0,3,4,0,0,4,9,2,0,0,5,16,3,1,0,0,6,25,12,17,0,0,0,7,36,10,58,
%T A108887 12,0,0,0,8,49,30,71,1,0,0,0,0,9,64,21,145,113,4,0,0,0,0,10,81,56,527,93,
%U A108887 47,1,0,0,0,0,11,100,36,294,235,21,127,0,0,0,0,0,12,121,90,347,202,199
%N A108887 Mirror image of triangle A108885.
%Y A108887 Adjacent sequences: A108884 A108885 A108886 this_sequence A108888 A108889 A108890
%Y A108887 Sequence in context: A113020 A127258 A049242 this_sequence A095884 A128908 A101603
%K A108887 nonn,tabl,frac
%O A108887 0,4
%A A108887 njas, Jul 16 2005
%I A095884
%S A095884 0,1,0,2,1,0,3,4,1,0,4,9,8,1,0,5,16,27,16,1,0,6,25,64,81,32,1,0,7,36,
%T A095884 125,256,243,64,1,0,8,49,216,625,1024,729,128,1,0,9,64,343,1296,3125,
%U A095884 4096,2187,256,1,0,10,81,512,2401,7776,15625,16384,6561,512,1,0,11,100
%N A095884 Triangle read by rows: T(n,k) = (n-k)^k, n>=1, 1<=k<=n.
%Y A095884 Adjacent sequences: A095881 A095882 A095883 this_sequence A095885 A095886 A095887
%Y A095884 Sequence in context: A127258 A049242 A108887 this_sequence A128908 A101603 A124030
%K A095884 easy,nonn,tabl
%O A095884 1,4
%A A095884 Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004
%I A128908
%S A128908 1,0,1,0,2,1,0,3,4,1,0,4,10,6,1,0,5,20,21,8,1,0,6,35,56,36,10,1,0,7,56,
%T A128908 126,120,55,12,1,0,8,84,252,330,220,78,14,1,0,9,120,462,792,715,364,105,
%U A128908 16,1,0,10,165,792,1716,2002,1365,560,136,18,1
%N A128908 Riordan array (1,x/(1-x)^2).
%C A128908 Triangle T(n,k), 0<=k<=n, read by rows given by [0,2,-1/2,1/2,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 .
%C A128908 Row sums give A088305 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007
%F A128908 T(n,0)=0^n, T(n,k)=binomial(n+k-1,2k-1) for k>=1 .
%e A128908 Triangle begins:
%e A128908 1;
%e A128908 0, 1;
%e A128908 0, 2, 1;
%e A128908 0, 3, 4, 1;
%e A128908 0, 4, 10, 6, 1;
%e A128908 0, 5, 20, 21, 8, 1;
%e A128908 0, 6, 35, 56, 36, 10, 1;
%e A128908 0, 7, 56, 126, 120, 55, 12, 1;
%e A128908 0, 8, 84, 252, 330, 220, 78, 14, 1;
%e A128908 0, 9, 120, 462, 792, 715, 364, 105, 16, 1;
%e A128908 0, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1 ;...
%Y A128908 Cf. A007318, A078812, A034008.
%Y A128908 Adjacent sequences: A128905 A128906 A128907 this_sequence A128909 A128910 A128911
%Y A128908 Sequence in context: A049242 A108887 A095884 this_sequence A101603 A124030 A106378
%K A128908 nonn,tabl
%O A128908 0,5
%A A128908 Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 22 2007
%I A101603
%S A101603 1,0,1,1,2,1,0,3,4,1,1,4,9,6,1,0,5,16,19,8,1,1,6,25,44,33,10,1,0,7,36,
%T A101603 85,96,51,12,1,1,8,49,146,225,180,73,14,1,0,9,64,231,456,501,304,99,16,
%U A101603 1,1,10,81,344,833,1182,985,476,129,18,1,0,11,100,489,1408,2471,2668
%N A101603 Riordan array (1/(1-x^2),x(1+x)/(1-x)).
%C A101603 Row sums are A097076(n+1). Diagonal sums are abs(A077902).
%F A101603 Number triangle whose columns are generated by x^k*(1+x)^(k-1)/(1-x)^(k+1)
%F A101603 Number triangle T(n, k)=sum{j=0..n-k, C(k-1, j)C(n-j, n-k-j)};
%e A101603 Rows start {1}, {0,1}, {1,2,1}, {0,3,4,1}, {1,4,9,6,1},...
%Y A101603 Adjacent sequences: A101600 A101601 A101602 this_sequence A101604 A101605 A101606
%Y A101603 Sequence in context: A108887 A095884 A128908 this_sequence A124030 A106378 A094301
%K A101603 easy,nonn,tabl
%O A101603 0,5
%A A101603 Paul Barry (pbarry(AT)wit.ie), Dec 08 2004
%I A124030
%S A124030 1,1,1,0,2,1,0,3,4,1,3,14,19,8,1,48,173,204,89,16,1,1505,4866,5173,2082,381,
%T A124030 32,1,108780,325990,316978,113481,18926,1580,64,1,19072536,53887686,48428411,
%U A124030 15201276,2206536,164222,6469,128,1,8332293760,22465873081,18859204368,5176293234
%V A124030 1,1,-1,0,-2,1,0,-3,4,-1,3,-14,19,-8,1,48,-173,204,-89,16,-1,1505,-4866,5173,-2082,381,
%W A124030 -32,1,108780,-325990,316978,-113481,18926,-1580,64,-1,19072536,-53887686,48428411,
%X A124030 -15201276,2206536,-164222,6469,-128,1,8332293760,-22465873081,18859204368,-5176293234
%N A124030 Binomial centered tridigonal matrices as a triangular sequence: t(n,m.d)=If[n + m - 1 == d, binomial[d - 1, n - 1], If[n + m == d, -1, If[n + m - 2 == d, -1, 0]]].
%C A124030 These are pretty matrices in terms of symmetry. Matrices: 1 X 1 {{1}} 2 X 2 {{1, -1}, {-1, 1}} 3 X 3 {{1, -1, 0}, {-1, 2, -1}, {0, -1, 1}} 4 X 4 {{1, -1, 0, 0}, {-1, 3, -1, 0}, {0, -1, 3, -1}, {0, 0, -1, 1}} 5 X 5 {{1, -1, 0, 0, 0}, {-1, 4, -1, 0, 0}, {0, -1, 6, -1, 0}, {0, 0, -1, 4, -1}, {0, 0, 0, -1, 1}} 6 X 6 {{1, -1, 0, 0, 0, 0}, {-1, 5, -1, 0, 0, 0}, {0, -1, 10, -1, 0, 0}, {0, 0, -1, 10, -1, 0}, {0, 0, 0, -1, 5, -1}, {0, 0, 0, 0, -1, 1}}
%F A124030 t(n,m.d)=If[n + m - 1 == d, binomial[d - 1, n - 1], If[n + m == d, -1, If[n + m - 2 == d, -1, 0]]]
%e A124030 Triangular sequence:
%e A124030 {1},
%e A124030 {1, -1},
%e A124030 {0, 2, 1},
%e A124030 {0, 3, 2, -1},
%e A124030 {3, -4, -11, 2, 1},
%e A124030 {48, -13, -106, 21, 6, -1},
%e A124030 {-1505, 36, 2693, -58, -129, 2, 1},
%e A124030 {-108780, 5530, 171342, -8705, -5290, 268,20, -1}
%t A124030 An[d_] := Table[If[n + m - 1 == d, Binomial[d - 1, n - 1], If[n + m ==d, -1, If[n + m - 2 == d, -1, 0]]], {n, 1, d}, {m, 1, d}]; Join[An[1], Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]
%Y A124030 Adjacent sequences: A124027 A124028 A124029 this_sequence A124031 A124032 A124033
%Y A124030 Sequence in context: A095884 A128908 A101603 this_sequence A106378 A094301 A135488
%K A124030 uned,probation,sign
%O A124030 1,5
%A A124030 Roger Bagula (rlbagulatftn(AT)yahoo.com), Nov 01 2006
%I A106378
%S A106378 1,2,1,0,3,4,1,5,6,1,12,16,13,0,14
%N A106378 Imaginary parts of numbers defined in A106377.
%Y A106378 Adjacent sequences: A106375 A106376 A106377 this_sequence A106379 A106380 A106381
%Y A106378 Sequence in context: A128908 A101603 A124030 this_sequence A094301 A135488 A099493
%K A106378 nonn
%O A106378 0,2
%A A106378 Sven Simon (sven-h.simon(AT)t-online.de), Apr 30 2005
%I A094301
%S A094301 0,1,2,1,0,3,4,1,6,21,16,31,456,681,1186,208441,671016,1869171,
%T A094301 43448848636,493709451721,3944060830326,1887802451244390439101,
%U A094301 245636825165950759470616,15799364856026522930297791
%N A094301 a(0)=0, a(1)=1, a(2)=2; for n>2, a(n) = a(n-1) - a(n-2) + a(n-3)^2.
%o A094301 (Perl) -e '@a=(0,1,2); for (3..20){ $a[$_] = $a[$_-1] - $a[$_-2] + $a[$_-3]**2; print "$a[$_],";}'
%o A094301 (PARI) print1(c=0,",",b=1,","a=2,",");for(n=1,21,print1(d=a-b+c^2,",");c=b;b=a;a=d) - Klaus Brockhaus
%Y A094301 Adjacent sequences: A094298 A094299 A094300 this_sequence A094302 A094303 A094304
%Y A094301 Sequence in context: A101603 A124030 A106378 this_sequence A135488 A099493 A088523
%K A094301 easy,nonn
%O A094301 0,3
%A A094301 Gamo (gamo(AT)telecable.es), Nov 13 2004
%E A094301 More terms from Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Nov 14 2004
%I A135488
%S A135488 1,1,1,0,1,2,1,0,3,4,3,0,5,8,15,0,17,48,27,0,63,96,89,0,205,320,513,
%T A135488 0,565,1920,961,0,3267,4352,4095,0,7085,13824,20475,0,25625,64512,
%U A135488 49923,0,184275,182272,178481,0,299593,839680
%N A135488 Number of distinct self-dual normal bases for GF(2^n) over GF(2)
%H A135488 Max Alekseyev, PARI scripts
%H A135488 Joerg Arndt, Fxtbook, see p. 883.
%H A135488 Dieter Jungnickel, Alfred J. Menezes and Scott A. Vanstone, On the Number of Self-Dual Bases of GF(q^m) Over GF(q), Proc. Amer. Math. Soc. 109 (1990), 23-29.
%Y A135488 Cf. A088437.
%Y A135488 Adjacent sequences: A135485 A135486 A135487 this_sequence A135489 A135490 A135491
%Y A135488 Sequence in context: A124030 A106378 A094301 this_sequence A099493 A088523 A035543
%K A135488 nonn
%O A135488 1,6
%A A135488 Max Alekseyev, Feb 11 2008
%I A099493
%S A099493 1,0,1,2,1,0,3,4,3,8,7,10,23,8,33,56,1,104,121,58,297,232,291,780,349,1072,
%T A099493 1903,174,3407,4272,1505,9840,8543,8752,26321,13902,33777,65456,11805,110356,
%U A099493 150173,35192,325303,310054,257319,885496,537919,1054888,2240927
%V A099493 1,0,1,2,-1,0,3,-4,-3,8,-7,-10,23,-8,-33,56,1,-104,121,58,-297,232,291,-780,349,1072,
%W A099493 -1903,174,3407,-4272,-1505,9840,-8543,-8752,26321,-13902,-33777,65456,-11805,-110356,
%X A099493 150173,35192,-325303,310054,257319,-885496,537919,1054888,-2240927
%N A099493 Expansion of (1+x^2)^2/(1+x^2-2x^3+x^4+x^6).
%C A099493 A Chebyshev transform of A052907, which has g.f. 1/(1-2x^2-2x^3). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).
%F A099493 a(n)=-a(n-2)+2a(n-3)-a(n-4)-a(n-6); a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^k*sum{j=0.., floor(n-2k/2), C(j, n-2k-2j)2^j}}.
%Y A099493 Adjacent sequences: A099490 A099491 A099492 this_sequence A099494 A099495 A099496
%Y A099493 Sequence in context: A106378 A094301 A135488 this_sequence A088523 A035543 A105546
%K A099493 easy,sign
%O A099493 0,4
%A A099493 Paul Barry (pbarry(AT)wit.ie), Oct 19 2004
%I A088523
%S A088523 2,1,0,3,4,5,1,4,0,9,7,8,10,11,13,2,10,17,8,19,8,21,12,5,2,25,20,15,8,1,
%T A088523 4,7,12,15,24,31,3,14,25,38,12,25,1,18,35,4,27,10,41,20,49,28,4,39,21,4,
%U A088523 45,26,8,49,27,10,2,57,45,32,28,25,27,26,24,23,25,28,32,35,39,46,52,61
%N A088523 a(1) = 2; for n > 1, a(n) = (a(n-1) + prime(n)) mod n.
%t A088523 a[1] = 2; a[n_] := a[n] = Mod[a[n - 1] + Prime[n], n]; Table[a[n], {n, 68, 81}]
%Y A088523 Cf. A088521, A088522, A006257.
%Y A088523 Adjacent sequences: A088520 A088521 A088522 this_sequence A088524 A088525 A088526
%Y A088523 Sequence in context: A094301 A135488 A099493 this_sequence A035543 A105546 A059297
%K A088523 nonn,easy
%O A088523 1,1
%A A088523 njas, Nov 14 2003
%E A088523 More terms from Robert G. Wilson v (rgwv(AT)rgwv.com) and Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 15 2003
%I A035543
%S A035543 1,0,0,2,1,0,3,5,2,4,9,11,8,13,25,27,22,39,61,59,60,102,136,131,154,236,
%T A035543 287,289,360,511,595,619,788,1062,1191,1286,1655,2113,2338,2608,3318,
%U A035543 4084,4504,5125,6441,7712,8502,9820,12156,14243,15763,18373,22388,25828
%N A035543 Number of partitions of n with equal number of parts congruent to each of 1 and 2 (mod 4)
%Y A035543 Adjacent sequences: A035540 A035541 A035542 this_sequence A035544 A035545 A035546
%Y A035543 Sequence in context: A135488 A099493 A088523 this_sequence A105546 A059297 A077874
%K A035543 nonn
%O A035543 0,4
%A A035543 Olivier Gerard (ogerard(AT)ext.jussieu.fr)
%E A035543 More terms from David W. Wilson (davidwwilson(AT)comcast.net)
%I A105546
%S A105546 2,1,0,3,5,9,7,4,9,6,3,3,9,8,9,7,2,6,2,6,1,9,9,3,9,6,4,9,6,8,5,3,2,5,4,
%T A105546 4,4,0,4,2,1,6,2,2,8,8,2,4,0,0,1,3,8,7,2,9,8,6,8,7,2,8,4,5,6,3,8,8,5,1,
%U A105546 7,0,8,4,8,3,7,3,6,2,3,2,1,8,4,6,6,9,7,4,7,6,3,3,5,5,2,1,9,4,4,9,4,0,9
%N A105546 Decimal expansion of prime nested radical.
%C A105546 A105547 is the continued fraction representation of this prime nested radical. A105548 is the similar semiprime nested radical. A105548 is the Fibonacci nested radical. Sqrt(1 + Sqrt(2 + Sqrt(3 + Sqrt(4 + ... = ~ 1.75793275661800... "It was discovered by T. Vijayaraghavan that the infinite radical, sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ... where a_n => 0, will converge to a limit if and only if the limit of (ln a_n)/2^n exists." [Clawson, 229; cf. A072449]. We know the asymptotic limit of primes, and hence that the Prime Nested Radical converges.
%D A105546 Borwein, J. M. and de Barra, G., Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.
%D A105546 Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 and 229.
%D A105546 Finch, S. R., Analysis of a Radical Expansion, Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.
%H A105546 Eric Weisstein's World of Mathematics, Nested Radical Constant.
%F A105546 Sqrt(2 + Sqrt(3 + Sqrt(5 + Sqrt(7 + Sqrt(11 + ... + Sqrt(Prime(n))))).
%e A105546 2.10359749633989726261993964968532544404216228824001387298687284563...
%t A105546 RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Prime[ Range[ 80]]]], 10, 111][[1]] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 31 2005)
%Y A105546 Cf. A000040, A072449.
%Y A105546 Adjacent sequences: A105543 A105544 A105545 this_sequence A105547 A105548 A105549
%Y A105546 Sequence in context: A099493 A088523 A035543 this_sequence A059297 A077874 A090683
%K A105546 cons,nonn
%O A105546 1,1
%A A105546 Jonathan Vos Post (jvospost2(AT)yahoo.com), Apr 12 2005
%I A059297
%S A059297 1,0,1,0,2,1,0,3,6,1,0,4,24,12,1,0,5,80,90,20,1,0,6,240,540,240,
%T A059297 30,1,0,7,672,2835,2240,525,42,1,0,8,1792,13608,17920,7000,1008,
%U A059297 56,1,0,9,4608,61236,129024,78750,18144,1764,72,1,0,10,11520,262440
%N A059297 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1.
%D A059297 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].
%H A059297 G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics.
%F A059297 E.g.f.: exp(x*y*exp(y)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Nov 18 2003
%e A059297 Triangle begins 1; 0, 1; 0, 2, 1; 0, 3, 6, 1; 0, 4, 24, 12, 1; ...
%Y A059297 There are 4 versions: A059297-A059300. Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums are A000248.
%Y A059297 Adjacent sequences: A059294 A059295 A059296 this_sequence A059298 A059299 A059300
%Y A059297 Sequence in context: A088523 A035543 A105546 this_sequence A077874 A090683 A117652
%K A059297 nonn,tabl
%O A059297 0,5
%A A059297 njas, Jan 25 2001
%I A077874
%S A077874 1,2,1,0,3,6,1,4,7,18,3,24,19,62,23,108,63,234,107,448,235,918,447,1812,
%T A077874 919,3650,1811,7272,3651,14574,7271,29116,14575,58266,29115,116496,58267,
%U A077874 233030,116495,466020,233031,932082,466019,1864120,932083,3728286,1864119
%V A077874 1,2,1,0,3,6,1,-4,7,18,-3,-24,19,62,-23,-108,63,234,-107,-448,235,918,-447,-1812,
%W A077874 919,3650,-1811,-7272,3651,14574,-7271,-29116,14575,58266,-29115,-116496,58267,
%X A077874 233030,-116495,-466020,233031,932082,-466019,-1864120,932083,3728286,-1864119
%N A077874 Expansion of (1-x)^(-1)/(1-x+2*x^2-2*x^3).
%Y A077874 Adjacent sequences: A077871 A077872 A077873 this_sequence A077875 A077876 A077877
%Y A077874 Sequence in context: A035543 A105546 A059297 this_sequence A090683 A117652 A095859
%K A077874 sign
%O A077874 0,2
%A A077874 njas, Nov 17 2002
%I A090683
%S A090683 1,0,1,0,2,1,0,3,9,1,0,4,42,24,1,0,5,150,250,50,1,0,6,465,1800,975,90,1,
%T A090683 0,7,1323,10535,12250,2940,147,1,0,8,3556,54096,119070,58800,7448,224,1,
%U A090683 0,9,9180,254100,979020,875826,222264,16632,324,1
%N A090683 Triangle read by rows defined by T(n,k)= A007318(n,k)*A048993(n,k).
%C A090683 {1}; {0, 1}; {0, 2, 1 }; {0, 3, 9, 1}; {0, 4, 42, 24, 1}; ...
%F A090683 T(n, k) = A090657(n, k)/k!.
%Y A090683 Cf. A07318 A048993 A090657.
%Y A090683 Adjacent sequences: A090680 A090681 A090682 this_sequence A090684 A090685 A090686
%Y A090683 Sequence in context: A105546 A059297 A077874 this_sequence A117652 A095859 A088850
%K A090683 easy,nonn,tabl
%O A090683 0,5
%A A090683 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 18 2003
%I A117652
%S A117652 0,0,1,1,2,1,0,3,10,20,35,55,84,120,168,227,300,388,495,621,770,943,
%T A117652 1144,1374,1638,1937,2275,2654,3080,3553,4080,4662,5304,6009,6783,7628,
%U A117652 8550,9552,10640,11817,13090,14462,15939,17525,19228,21050,23000,25081
%N A117652 A052472 for half integer values as might be found in fractal dimensional tensors.
%C A117652 Compare Magic numbers (A018226) 2, 8, 20, 28, 50, 82, 126, 168 to 3, 10, 20, 35, 55, 84, 120, 168
%F A117652 A052472[n]=n*(n + 1)*(n + 2)*(n - 3)/12 a(n) = Abs[Floor[A0542472[n]]]
%t A117652 f[n_] = n*(n + 1)*(n + 2)*(n - 3)/12 a = Table[Floor[Abs[f[n]]], {n, 0, 25, 1/2}]
%Y A117652 Cf. A052472, A018226.
%Y A117652 Adjacent sequences: A117649 A117650 A117651 this_sequence A117653 A117654 A117655
%Y A117652 Sequence in context: A059297 A077874 A090683 this_sequence A095859 A088850 A130125
%K A117652 nonn,uned,probation
%O A117652 0,5
%A A117652 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 11 2006
%I A095859
%S A095859 0,1,0,2,1,0,3,16,1,0,4,81,512,1,0,5,256,19683,65536,1,0,6,625,262144,
%T A095859 43046721,33554432,1,0,7,1296,1953125,4294967296,847288609443,
%U A095859 68719476736,1,0,8,2401,10077696,152587890625,1125899906842624
%N A095859 Triangle read by rows: T(n,k) = (n-k)^(k^2), n>=1, 1<=k<=n.
%Y A095859 Adjacent sequences: A095856 A095857 A095858 this_sequence A095860 A095861 A095862
%Y A095859 Sequence in context: A077874 A090683 A117652 this_sequence A088850 A130125 A137336
%K A095859 easy,nonn,tabl
%O A095859 1,4
%A A095859 Herman Jamke (hermanjamke(AT)fastmail.fm), Jul 10 2004
%I A088850
%S A088850 0,0,1,1,1,1,2,1,0,4,0,1,3,1,4,2
%N A088850 Number of different values of k when A056239(k) = A056239(k+1) = n.
%Y A088850 Adjacent sequences: A088847 A088848 A088849 this_sequence A088851 A088852 A088853
%Y A088850 Sequence in context: A090683 A117652 A095859 this_sequence A130125 A137336 A115322
%K A088850 easy,fini,more,nonn
%O A088850 0,7
%A A088850 Naohiro Nomoto (pcmusume(AT)m11.alpha-net.ne.jp), Nov 24 2003
%I A130125
%S A130125 1,0,2,1,0,4,0,2,0,8,1,0,4,0,16,0,2,0,8,0,32,1,0,4,0,16,0,64,0,2,0,8,0,
%T A130125 32,0,128,1,0,4,0,16,0,64,0,256
%N A130125 A128174 * A130123.
%C A130125 Row sums = A000975: (1, 2, 5, 10, 21, 42,...).
%F A130125 A128174 * A130123 as infinite lower triangular matrices. By columns, (2^k, 0, 2^k, 0,...)
%e A130125 First few rows of the triangle are:
%e A130125 1;
%e A130125 0, 2;
%e A130125 1, 0, 4;
%e A130125 0, 2, 0, 8;
%e A130125 1, 0, 4, 0, 16;
%e A130125 0, 2, 0, 8, 0, 32;
%e A130125 ...
%Y A130125 Cf. A000975, A128174, A130123.
%Y A130125 Adjacent sequences: A130122 A130123 A130124 this_sequence A130126 A130127 A130128
%Y A130125 Sequence in context: A117652 A095859 A088850 this_sequence A137336 A115322 A053117
%K A130125 nonn,tabl
%O A130125 0,3
%A A130125 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 11 2007
%I A137336
%S A137336 0,0,2,1,0,4,0,4,0,8,1,0,12,0,16,0,6,0,32,0,32,1,0,24,0,80,0,64,0,8,0,
%T A137336 80,0,192,0,128,1,0,40,0,240,0,448,0,256,0,10,0,160,0,672,0,1024,0,512,
%U A137336 1,0,60,0,560,0,1792,0,2304,0,1024
%V A137336 0,0,-2,1,0,-4,0,4,0,-8,-1,0,12,0,-16,0,-6,0,32,0,-32,1,0,-24,0,80,0,-64,0,8,0,-80,0,
%W A137336 192,0,-128,-1,0,40,0,-240,0,448,0,-256,0,-10,0,160,0,-672,0,1024,0,-512,1,0,-60,0,560,
%X A137336 0,-1792,0,2304,0,-1024
%N A137336 Triangular sequence of coefficients from expansion of the U(x,n) one's opposite expansion: 1/(1-2*xt+t^2)=Sum[U(x,n),{n,0,Infinity}]; 1/(1-2*xt+t^2)+(-2*x*t+t^1)/(1-2*xt+t^2)=1 so that: (-2*x*t+t^1)/(1-2*xt+t^2)=Sum[p(x,n),{n,0,Infinity}].
%C A137336 Row sums are:
%C A137336 {0, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11};
%C A137336 This polynomial set appears to be a Chebyshev related set of polynomials.
%C A137336 This sequence was suggested by the fact that except for zero the expansion of
%C A137336 f(x,t)=(-x*t+t^2)/(1-2*x*t+t^2) =Sum[q(x,n),{n,0,Infinity}]
%C A137336 is
%C A137336 q(x,n)= -T(x,n).
%C A137336 Integration of the recursive polynomials shows alternating orthogonality:
%C A137336 Table[Integrate[p[x, n]*p[x, m]/Sqrt[1 - x^2], {x, -1, 1}], {n, 0, 10}, {m, 0, 10}]
%D A137336 Rosenblum and Rovnyak, Hardy Classes and Operator Theory,Dover, New York,1985, page 18-19
%F A137336 Expansion of (-2*x*t+t^1)/(1-2*xt+t^2)
%e A137336 {0},
%e A137336 {0, -2},
%e A137336 {1, 0, -4},
%e A137336 {0, 4, 0, -8},
%e A137336 {-1, 0, 12, 0, -16},
%e A137336 {0, -6, 0, 32,0, -32},
%e A137336 {1, 0, -24, 0, 80, 0, -64},
%e A137336 {0, 8, 0, -80, 0, 192,0, -128},
%e A137336 {-1, 0, 40, 0, -240, 0, 448, 0, -256},
%e A137336 {0, -10, 0, 160, 0, -672, 0, 1024, 0, -512},
%e A137336 {1, 0, -60, 0, 560, 0, -1792, 0, 2304, 0, -1024}
%t A137336 Clear[p] p[t_] = (-2*x*t + t^2)/(1 - 2*x*t + t^2); Table[ ExpandAll[SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Join[{{0}}, Table[ CoefficientList[ExpandAll[SeriesCoefficient[Series[p[t], {t, 0, 30}], n]], x], {n, 0, 10}]]; Flatten[a] (* polynomial recursion: needs first three terms*) Clear[p] p[x, 0] = 0; p[x, 1] = -2*x; p[x, 2] = 1 - 4*x^2; p[x_, n_] := p[x, n] = 2*x*p[x, n - 1] - p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, Length[g] - 1}]
%Y A137336 Adjacent sequences: A137333 A137334 A137335 this_sequence A137337 A137338 A137339
%Y A137336 Sequence in context: A095859 A088850 A130125 this_sequence A115322 A053117 A121448
%K A137336 nonn,uned,tabl
%O A137336 1,3
%A A137336 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 07 2008
%I A115322
%S A115322 1,0,2,1,0,4,0,4,0,8,1,0,12,0,16,0,6,0,32,0,32,1,0,24,0,80,0,64,0,8,0,
%T A115322 80,0,192,0,128,1,0,40,0,240,0,448,0,256,0,10,0,160,0,672,0,1024,0,512,
%U A115322 1,0,60,0,560,0,1792,0,2304,0,1024,0,12,0,280,0,1792,0,4608,0,5120,0
%N A115322 Triangle of coefficients of Pell polynomials.
%C A115322 Aside from signs, same as A053117.
%H A115322 Eric Weisstein's World of Mathematics, Pell Polynomial
%F A115322 Fibonacci[n, 2x].
%e A115322 1, 2*x, 1 + 4*x^2, 4*x + 8*x^3, 1 + 12*x^2 + 16*x^4, ...
%Y A115322 Cf. A053117.
%Y A115322 Adjacent sequences: A115319 A115320 A115321 this_sequence A115323 A115324 A115325
%Y A115322 Sequence in context: A088850 A130125 A137336 this_sequence A053117 A121448 A019094
%K A115322 nonn,tabl
%O A115322 1,3
%A A115322 Eric Weisstein (eric(AT)weisstein.com), Jan 20, 2006
%I A053117
%S A053117 1,0,2,1,0,4,0,4,0,8,1,0,12,0,16,0,6,0,32,0,32,1,0,24,0,80,
%T A053117 0,64,0,8,0,80,0,192,0,128,1,0,40,0,240,0,448,0,256,0,10,0,
%U A053117 160,0,672,0,1024,0,512,1,0,60,0,560,0,1792,0,2304,0,1024,0,12,0,280,0,1792,0,4608,0,5120,0,2048,1,0,84,0,1120,0,5376,0,11520
%V A053117 1,0,2,-1,0,4,0,-4,0,8,1,0,-12,0,16,0,6,0,-32,0,32,-1,0,24,0,-80,
%W A053117 0,64,0,-8,0,80,0,-192,0,128,1,0,-40,0,240,0,-448,0,256,0,10,0,
%X A053117 -160,0,672,0,-1024,0,512,-1,0,60,0,-560,0,1792,0,-2304,0,1024,0,-12,0,280,0,-1792,0,4608,0,-5120,0,2048,1,0,-84,0,1120,0,-5376,0,11520
%N A053117 Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).
%C A053117 a(n,m)= (2^m)*A049310(n,m).
%C A053117 G.f. for row polynomials U(n,x) (signed triangle): 1/(1-2*x*z+z^2). Unsigned triangle |a(n,m)| has Fibonacci polynomials F(n+1,2*x) as row polynomials with G.f. 1/(1-2*x*z-z^2).
%C A053117 Row sums (unsigned triangle) A000129(n+1) (Pell). Row sums (signed triangle) A000027(n+1) (natural numbers).
%D A053117 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
%D A053117 A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
%H A053117 T. D. Noe, Rows n=0..100 of triangle, flattened
%H A053117 R. Pemantle and M. C. Wilson, Asymptotics of multivariate sequences, I: smooth points of the singular variety
%H A053117 Index entries for sequences related to Chebyshev polynomials.
%F A053117 a(n, m) := 0 if n=0, k>=0). A binary tree is a rooted tree in which each vertex has at most two children and each child of a vertex is designated as its left or right child.
%C A121448 T(2n,0)=binomial(2n,n)/(n+1) (the Catalan numbers; A000108); T(2n+1,0)=0. T(n,n)=2^n (A000079). Sum(k*T(n,k),k=0..n)=2*binomial(2n,n-1)=2*A001791(n). After deleting the zeros, reflection of A091894.
%F A121448 T(n,k)=2^k*binomial(n+1,k)binomial(n+1-k,(n-k)/2)/(n+1) if n-k is even; otherwise, T(n,k)=0. G.f. G=G(t,z) satisfies G=1+2tzG+z^2*G^2.
%F A121448 T(n,k)=2^k*A097610(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 17 2006
%F A121448 T(n,k) = A097610(n,k)*2^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 22 2006
%e A121448 T(2,2)=4 because, denoting by L (R) an edge going from a vertex to a left (right) child, we have the paths: LL, LR, RL, and RR.
%e A121448 Triangle starts:
%e A121448 1;
%e A121448 0,2;
%e A121448 1,0,4;
%e A121448 0,6,0,8;
%e A121448 2,0,24,0,16;
%p A121448 T:=proc(n,k) if n-k mod 2 = 0 then 2^k*binomial(n+1,k)*binomial(n+1-k,(n-k)/2)/(n+1) else 0 fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%Y A121448 Cf. A000108, A000079, A001791, A091894.
%Y A121448 Adjacent sequences: A121445 A121446 A121447 this_sequence A121449 A121450 A121451
%Y A121448 Sequence in context: A137336 A115322 A053117 this_sequence A019094 A134082 A136329
%K A121448 nonn,tabl
%O A121448 0,3
%A A121448 Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 31 2006
%I A019094
%S A019094 0,2,1,0,4,0,8,4,47,76,277,596,1637,3606,8987,19704,50026,110922
%N A019094 Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite CHI = Chiavennite Ca4Mn4 [ Be8Si20O52(OH)8 ] . 8 H2O.
%D A019094 G. Thimm and W. E. Klee, Zeolite cycle sequences, Zeolites, 19, pp. 422-424, 1997.
%H A019094 G. Thimm, Cycle sequences of crystal structures
%Y A019094 Adjacent sequences: A019091 A019092 A019093 this_sequence A019095 A019096 A019097
%Y A019094 Sequence in context: A115322 A053117 A121448 this_sequence A134082 A136329 A122073
%K A019094 nonn
%O A019094 3,2
%A A019094 Georg Thimm (mgeorg(AT)ntu.edu.sg)
%I A134082
%S A134082 1,2,1,0,4,1,0,0,6,1,0,0,0,8,1,0,0,0,0,10,1,0,0,0,0,0,12,1,0,0,0,0,0,0,
%T A134082 14,1
%N A134082 Triangle read by rows, (n-1) zeros followed by (2n, 1).
%C A134082 Row sums = (1, 3, 5, 7,...). A134082 * [1,2,3,...] = A084849: (1, 4, 11, 22, 37,...). Binomial transform of A134082 = A134083. A112295 replaces subdiagonal with (-1,-3,-5,...)
%F A134082 Triangle read by rows, (n-1) zeros followed by (2n, 1). As an infinite lower triangular matrix, (1,1,1,...) in the main diagonal and (2,4,6,8,...) in the subdiagonal.
%F A134082 From formalism in A132382, e.g.f. = I_o[2*(u*x)^(1/2)] (1+2x) where I_o is the zeroth modified Bessel function of the first kind, i.e. I_o[2*(u*x)^(1/2)] = sum(j=0,1,...) u^j/j! * x^j/j! . - Tom Copeland (tcjpn(AT)msn.com), Dec 07 2007
%e A134082 First few rows of the triangle are:
%e A134082 1;
%e A134082 2, 1;
%e A134082 0, 4, 1;
%e A134082 0, 0, 6, 1;
%e A134082 0, 0, 0, 8, 1;
%e A134082 0, 0, 0, 0, 10, 1;
%e A134082 ...
%Y A134082 Cf. A112295, A084849, A134083.
%Y A134082 Adjacent sequences: A134079 A134080 A134081 this_sequence A134083 A134084 A134085
%Y A134082 Sequence in context: A053117 A121448 A019094 this_sequence A136329 A122073 A106236
%K A134082 nonn,tabl
%O A134082 0,2
%A A134082 Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 07 2007
%I A136329
%S A136329 1,2,1,0,4,1,2,7,6,1,4,8,18,8,1,6,5,38,33,10,1,8,4,63,96,52,12,1,10,21,
%T A136329 84,222,190,75,14,1,12,48,84,432,550,328,102,16,1,14,87,36,726,1342,
%U A136329 1131,518,133,18,1,16,140,99,1056,2860,3276,2065,768,168,20,1
%V A136329 1,-2,1,0,-4,1,2,7,-6,1,-4,-8,18,-8,1,6,5,-38,33,-10,1,-8,4,63,-96,52,-12,1,10,-21,-84,
%W A136329 222,-190,75,-14,1,-12,48,84,-432,550,-328,102,-16,1,14,-87,-36,726,-1342,1131,-518,
%X A136329 133,-18,1,-16,140,-99,-1056,2860,-3276,2065,-768,168,-20,1
%N A136329 Triangular sequence of coefficients of a polynomial recursion for C_n and B_n Cartan matrices: p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2) p(x,n)=x2-4*x+4-m:m=4;(related sequence: A_n:m=1,G_n,m=3,B_n,C_n,m=2) This triangular sequence is an extension to the Cartan pattern of matrices.
%C A136329 Row sums are:
%C A136329 {1, -1, -3, 4, -1, -3, 4, -1, -3, 4, -1}
%C A136329 This sequence is also related to different p(x,2) start:
%C A136329 1) A_n like sequence A053122 ( sign change)
%C A136329 2) my G_n matrix A136674
%C A136329 3) B_n,C_n A110162
%F A136329 p(x, n) = (-2 + x)*p(x, n - 1) - p(x, n - 2) Three start vectors necessary: p(x,0)=1;p(x,1)=2-x; p(x,2)=x^2-4*x=CharacteristicPolynomial[{{2, -4}, {-1, 2}}, x] or CharacteristicPolynomial[{{2, -1}, {-4, 2}}, x]
%e A136329 {1},
%e A136329 {-2, 1},
%e A136329 {0, -4, 1},
%e A136329 {2, 7, -6, 1},
%e A136329 {-4, -8, 18, -8, 1},
%e A136329 {6, 5, -38, 33, -10,1},
%e A136329 {-8, 4, 63, -96, 52, -12, 1},
%e A136329 {10, -21, -84, 222, -190, 75, -14, 1},
%e A136329 {-12, 48, 84, -432, 550, -328, 102, -16, 1},
%e A136329 {14, -87, -36, 726, -1342, 1131, -518, 133, -18, 1},
%e A136329 {-16, 140, -99, -1056, 2860, -3276, 2065, -768, 168, -20, 1}
%t A136329 Clear[p, a] p[x, 0] = 1; p[x, 1] = -2 + x; p[x, 2] = x^2 - 4*x ; p[x_, n_] := p[x, n] = (-2 + x)*p[x, n - 1] - p[x, n - 2]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}] Flatten[a]
%Y A136329 Cf. A053122, A136674, A110162.
%Y A136329 Adjacent sequences: A136326 A136327 A136328 this_sequence A136330 A136331 A136332
%Y A136329 Sequence in context: A121448 A019094 A134082 this_sequence A122073 A106236 A122792
%K A136329 nonn,tabl,uned
%O A136329 1,2
%A A136329 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 12 2008
%I A122073
%S A122073 1,2,1,0,4,1,2,9,8,1,2,3,19,12,1,4,6,47,55,18,1,2,15,0,88,93,24,1,2,23,7,
%T A122073 190,324,182,32,1,0,12,63,62,332,554,274,40,1,2,9,108,133,678,1642,1346,450,
%U A122073 50,1,2,11,55,276,463,1129,2832,2128,630,60,1,4,30,71,543,1044,2204,7761
%V A122073 1,2,-1,0,-4,1,2,-9,8,-1,-2,-3,19,-12,1,-4,-6,47,-55,18,-1,2,15,0,-88,93,-24,1,2,23,-7,
%W A122073 -190,324,-182,32,-1,0,-12,-63,62,332,-554,274,-40,1,2,-9,-108,133,678,-1642,1346,-450,
%X A122073 50,-1,-2,-11,55,276,-463,-1129,2832,-2128,630,-60,1,-4,-30,71,543,-1044,-2204,7761
%N A122073 Triangular array from Steinbach matrices plus their squares as characteristic polynomials: M[i,j]=A[i,j]+A[i,j]^2: A[i,j]^2=Min[i,j]; CharacteristicPolynomial[M[i,j]];.
%C A122073 Based on the idea that the Steinbach matrices form a "golden Field". Matrices are: {{2, 2}, {2, 2}}, {{2, 2, 2}, {2, 3, 2}, {2, 2, 3}}, {{2, 2, 2, 2}, {2, 3, 3, 2}, {2, 3, 3, 3}, {2, 2, 3, 4}}, {{2, 2, 2, 2, 2}, {2, 3, 3, 3, 2}, {2, 3, 4, 3, 3}, {2, 3, 3, 4, 4}, {2, 2, 3, 4, 5}}, {{2, 2, 2, 2, 2, 2}, {2,3, 3, 3, 3, 2}, {2, 3, 4, 4, 3, 3}, {2, 3, 4, 4, 4, 4}, {2, 3, 3, 4, 5, 5}, {2, 2, 3, 4, 5, 6}}
%D A122073 Peter Steinbach, "Golden Fields: A Case for the Heptagon", Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
%F A122073 dth level M(i,j)->An[d]; T(n,m)=CoefficientList[CharacteristicPolynomial[An[d], x], x]
%e A122073 {1},
%e A122073 {2, -1},
%e A122073 {0, -4, 1},
%e A122073 {2, -9, 8, -1},
%e A122073 {-2, -3, 19, -12, 1},
%e A122073 {-4, -6,47, -55, 18, -1}
%e A122073 {2, 15, 0, -88, 93, -24, 1},
%e A122073 {2, 23, -7, -190, 324, -182, 32, -1},
%e A122073 {0, -12, -63, 62, 332, -554, 274, -40, 1}
%t A122073 An[d_] := Table[Min[n, m] + If[n + m - 1 > d, 0, 1], {n, 1, d}, {m, 1, d}]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d,1, 20}]]; Flatten[%]
%Y A122073 Cf. A038223, A122599, A076756, A054142.
%Y A122073 Adjacent sequences: A122070 A122071 A122072 this_sequence A122074 A122075 A122076
%Y A122073 Sequence in context: A019094 A134082 A136329 this_sequence A106236 A122792 A139136
%K A122073 tabl,uned,sign
%O A122073 1,2
%A A122073 Gary Adamson (qntmpkt(AT)yahoo.com), Oct 16 2006
%I A106236
%S A106236 1,1,0,2,1,0,4,2,0,0,9,6,0,0,0,20,13,2,0,0,0,48,37,4,0,0,0,0,115,86,17,
%T A106236 0,0,0,0,0,286,239,46,0,0,0,0,0,0,719,577,142,8,0,0,0,0,0,0,1842,1607,
%U A106236 367,18,0,0,0,0,0,0,0,4766,4025,1136,76,0,0,0,0,0,0,0,0,12486,11185
%N A106236 Triangle of the numbers of different forests with m rooted trees having distinct orders.
%C A106236 a(n) = 0 if and only if n < m + (((1+m)*m - 1)^2 -1)/8, where m is the number of trees in the forests counted by a(n).
%F A106236 a(n)= sum over the partitions of N:1K1+2K2+ ... +NKN, with exactly m distinct parts, of product_{1=0} (1+y*A000081(k)*x^k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 14 2005
%e A106236 a(3)=0 because m = 2, and (see comments) 3 < (2 + 3).
%e A106236 a(4)>0 because m = 1. Note that (((1+m)*m - 1)^2 -1)/8 = 0, if m = 1. It is clear that n >= m.
%Y A106236 Cf. A106234, A000081.
%Y A106236 Adjacent sequences: A106233 A106234 A106235 this_sequence A106237 A106238 A106239
%Y A106236 Sequence in context: A134082 A136329 A122073 this_sequence A122792 A139136 A138002
%K A106236 nonn,tabl
%O A106236 1,4
%A A106236 Washington Bomfim (webonfim(AT)bol.com.br), Apr 28 2005
%I A122792
%S A122792 1,1,0,2,1,0,4,2,0,6,4,0,10,6,0,16,9,0,24,14,0,36,20,0,52,29,0,74,42,0,
%T A122792 104,58,0,144,80,0,198,110,0,268,148,0,360,198,0,480,264,0,634,347,0,
%U A122792 832,454,0,1084,592,0,1404,764,0,1808,982,0,2316,1257,0,2952,1598,0
%N A122792 Expansion of eta(q^2)^2/(eta(q)eta(q^3)) in powers of q.
%F A122792 Euler transform of period 6 sequence [ 1, -1, 2, -1, 1, 0, ...].
%F A122792 G.f.: Product_{k>0} (1-x^k)^2/(1+x^k+x^(2k)). a(3n+2)=0.
%o A122792 (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^2/eta(x+A)/eta(x^3+A), n))}
%Y A122792 A098151(n)=a(3n). A097197(n)=a(3n+1).
%Y A122792 Adjacent sequences: A122789 A122790 A122791 this_sequence A122793 A122794 A122795
%Y A122792 Sequence in context: A136329 A122073 A106236 this_sequence A139136 A138002 A062296
%K A122792 nonn
%O A122792 0,4
%A A122792 Michael Somos, Sep 11 2006
%I A139136
%S A139136 1,1,0,2,1,0,4,2,0,6,4,0,10,6,0,16,9,0,24,14,0,36,20,0,52,29,0,74,42,0,
%T A139136 104,58,0,144,80,0,198,110,0,268,148,0,360,198,0,480,264,0,634,347,0,
%U A139136 832,454,0,1084,592,0,1404,764,0,1808,982,0,2316,1257,0,2952,1598,0
%V A139136 1,-1,0,-2,1,0,4,-2,0,-6,4,0,10,-6,0,-16,9,0,24,-14,0,-36,20,0,52,-29,0,-74,42,0,104,
%W A139136 -58,0,-144,80,0,198,-110,0,-268,148,0,360,-198,0,-480,264,0,634,-347,0,-832,454,0,
%X A139136 1084,-592,0,-1404,764,0,1808,-982,0,-2316,1257,0,2952,-1598,0,-3744
%N A139136 Expansion of psi(-q) / f(q^3) where psi(), f() are Ramanujan theta functions.
%F A139136 Expansion of eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2) * eta(q^6)^3) in powers of q.
%F A139136 Euler transform of period 12 sequence [ -1, 0, -2, -1, -1, 2, -1, -1, -2, 0, -1, 0, ...].
%F A139136 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(-1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A139135.
%F A139136 a(3*n + 2) = 0.
%F A139136 G.f.: Product_{k>0} P(12, x^k) / ( (1 + x^(2*k-1))^2 * P(3, x^k) * P(6, x^k)^2) where P(n, x) is nth cyclotomic polynomial.
%e A139136 1 - q - 2*q^3 + q^4 + 4*q^6 - 2*q^7 - 6*q^9 + 4*q^10 + 10*q^12 - 6*q^13 + ...
%o A139136 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A) * eta(x^6 + A)^3), n))}
%Y A139136 A132002(n) = a(3*n). - A139135(n) = a(3*n + 1).
%Y A139136 Adjacent sequences: A139133 A139134 A139135 this_sequence A139137 A139138 A139139
%Y A139136 Sequence in context: A122073 A106236 A122792 this_sequence A138002 A062296 A091453
%K A139136 sign
%O A139136 0,4
%A A139136 Michael Somos, Apr 10 2008
%I A138002
%S A138002 0,0,0,2,1,0,4,2,0,6,5,4,8,5,2,10,5,0,12,10,8,14,9,4,16,8,0,18,15,12,20,
%T A138002 15,10,22,15,8,24,20,16,26,18,10,28,16,4,30,25,20,32,21,10,34,17,0,36,
%U A138002 30,24,38,29,20,40,28,16,42,35,28,44,31,18,46,27,8,48,40,32,50,33,16,52
%N A138002 a(3*n)=2*n, a(3*n+1)=a(n)+n, a(3*n+2)=2*a(n), a(n)=0 for n<3.
%C A138002 a(A062318(n)) = 0.
%H A138002 R. Zumkeller, Table of n, a(n) for n = 1..10000
%F A138002 a(n) = (n mod 3) * a(floor(n/3)) + (2 - n mod 3) * floor(n/3), a(n) = 0.
%Y A138002 Cf. A025480.
%Y A138002 Adjacent sequences: A137999 A138000 A138001 this_sequence A138003 A138004 A138005
%Y A138002 Sequence in context: A106236 A122792 A139136 this_sequence A062296 A091453 A062173
%K A138002 nonn
%O A138002 0,4
%A A138002 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 26 2008
%I A062296
%S A062296 0,0,0,2,1,0,4,2,0,8,7,6,9,6,3,10,5,0,16,14,12,16,11,6,16,8,0,26,25,24,
%T A062296 27,24,21,28,23,18,33,30,27,32,25,18,31,20,9,40,35,30,37,26,15,34,17,0,
%U A062296 52,50,48,52,47,42,52,44,36,58,53,48,55,44,33,52,35,18,64,56,48,58,41
%N A062296 Number of entries in n-th row of Pascal's triangle divisible by 3.
%F A062296 a(n) + A006047(n) = n + 1 so a(n) = n + 1 - A006047(n)
%e A062296 When n=3 the row is 1,3,3,1 so a(3) = 2.
%p A062296 p:=proc(n) local ct, k: ct:=0: for k from 0 to n do if binomial(n,k) mod 3 = 0 then else ct:=ct+1 fi od: end: seq(n+1-p(n),n=0..83); (Deutsch)
%Y A062296 Cf. A006047.
%Y A062296 Adjacent sequences: A062293 A062294 A062295 this_sequence A062297 A062298 A062299
%Y A062296 Sequence in context: A122792 A139136 A138002 this_sequence A091453 A062173 A004558
%K A062296 nonn
%O A062296 0,4
%A A062296 Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 02 2001
%E A062296 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 03 2005
%I A091453
%S A091453 0,0,2,1,0,4,2,1,1,0,6,3,2,1,1,1,0,8,4,2,2,1,1,1,1,0,10,5,3,2,2,1,1,1,1,
%T A091453 1,0,12,6,4,3,2,2,1,1,1,1,1,1,0,14,7,4,3,2,2,2,1,1,1,1,1,1,1,0,16,8,5,4,
%U A091453 3,2,2,2,1,1,1,1,1,1,1,1,0,18,9,6,4,3,3,2,2,2,1,1,1,1,1,1,1,1,0,20,10,6
%N A091453 Triangular array T(h,k) read by rows in which row h consists of the numbers Floor[2h/k], k=1,2,...,2h+1.
%C A091453 First terms in period of continued fraction of sqrt(n).
%C A091453 Similar to A013942, except that here, period lengths of continued fractions all sqrt(n) are included, those of the squares being 0.
%e A091453 Northwest corner of array:
%e A091453 0
%e A091453 0 2 1
%e A091453 0 4 2 1 1
%e A091453 0 6 3 2 1 1 1
%e A091453 0 8 4 2 2 1 1 1 1
%Y A091453 Cf. A013942, A091449.
%Y A091453 Adjacent sequences: A091450 A091451 A091452 this_sequence A091454 A091455 A091456
%Y A091453 Sequence in context: A139136 A138002 A062296 this_sequence A062173 A004558 A002349
%K A091453 nonn,tabf
%O A091453 1,3
%A A091453 Clark Kimberling (ck6(AT)evansville.edu), Feb 03 2004
%I A062173
%S A062173 0,0,1,0,1,2,1,0,4,2,1,8,1,2,4,0,1,14,1,8,4,2,1,8,16,2,13,8,1,2,1,0,4,
%T A062173 2,9,32,1,2,4,8,1,32,1,8,31,2,1,32,15,12,4,8,1,14,49,16,4,2,1,8,1,2,4,
%U A062173 0,16,32,1,8,4,22,1,32,1,2,34,8,9,32,1,48,40,2,1,32,16,2,4,40,1,32,64
%N A062173 2^(n-1) mod n.
%C A062173 If p is an odd prime then a(p)=1. However, a(n) is also 1 for pseudoprimes to base 2 such as 341.
%H A062173 Index entries for sequences related to pseudoprimes
%e A062173 a(5) = 2^(5-1) mod 5 = 16 mod 5 = 1.
%Y A062173 Cf. A001567, A015919, A062172.
%Y A062173 Adjacent sequences: A062170 A062171 A062172 this_sequence A062174 A062175 A062176
%Y A062173 Sequence in context: A138002 A062296 A091453 this_sequence A004558 A002349 A096794
%K A062173 nonn
%O A062173 1,6
%A A062173 Henry Bottomley (se16(AT)btinternet.com), Jun 12 2001
%I A004558
%S A004558 2,1,0,4,2,2,3,2,4,0,1,1,3,2,4,1,0,4,0,0,1,3,4,4,1,2,3,3,0,4,1,3,0,
%T A004558 4,2,4,2,2,2,1,2,1,3,2,1,1,3,0,1,3,1,0,3,2,1,0,0,1,0,2,2,1,4,2,3,4,
%U A004558 4,4,3,4,3,4,2,4,2,3,4,4,1,4,4,4,2,1,1,3,0,1,4,2,0,4,0,3,0,0,1,0,4
%N A004558 Expansion of sqrt(5) in base 5.
%Y A004558 Adjacent sequences: A004555 A004556 A004557 this_sequence A004559 A004560 A004561
%Y A004558 Sequence in context: A062296 A091453 A062173 this_sequence A002349 A096794 A106375
%K A004558 nonn,base,cons
%O A004558 1,1
%A A004558 njas
%I A002349 M0046 N0015
%S A002349 0,2,1,0,4,2,3,1,0,6,3,2,180,4,1,0,8,4,39,2,12,42,5,1,0,10,5,24,1820,2,
%T A002349 273,3,4,6,1,0,12,6,4,3,320,2,531,30,24,3588,7,1,0,14,7,90,9100,66,12,
%U A002349 2,20,2574,69,4,226153980,8,1,0,16,8,5967,4,936,30,413,2,267000,430,3
%N A002349 Take solution to Pellian equation x^2 - n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is a square. A002350 gives values of x.
%D A002349 A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
%D A002349 C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
%D A002349 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
%D A002349 Albert H. Beiler, "The Pellian" (chap 22), Recreations in the Theory of Numbers, 2nd ed. NY: Dover, 1966.
%D A002349 E. E. Whitford, The Pell Equation.
%H A002349 T. D. Noe, Table of n, a(n) for n=1..1000
%H A002349 E. E. Whitford, The Pell equation, New York, 1912.
%H A002349 L. Euler, De solutione problematum diophanteorum per numeros integros, par. 17
%e A002349 For n = 1, 2, 3, 4, 5 solutions are (x,y) = (1, 0), (3, 2), (2, 1), (1, 0), (9, 4).
%t A002349 a[n_] := If[IntegerQ[Sqrt[n]], 0, For[y=1, !IntegerQ[Sqrt[n*y^2+1]], y++, Null]; y]
%t A002349 PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cof = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cof]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; f[n_] := If[ !IntegerQ[ Sqrt[n]], PellSolve[n][[2]], 0]; Table[ f[n], {n, 0, 75}]
%Y A002349 Cf. A002350, A006702, A006703, A006704, A006705. See A033316, A033315, A033319 for records.
%Y A002349 Adjacent sequences: A002346 A002347 A002348 this_sequence A002350 A002351 A002352
%Y A002349 Sequence in context: A091453 A062173 A004558 this_sequence A096794 A106375 A131667
%K A002349 nonn,nice,easy
%O A002349 1,2
%A A002349 njas
%E A002349 More terms from Enoch Haga (Enokh(AT)comcast.net), Mar 14 2002. Better description from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 14 2003
%I A096794
%S A096794 1,0,2,1,0,4,2,4,0,8,6,8,12,0,16,18,26,24,32,0,32,57,80,84,64,80,0,64,
%T A096794 186,260,264,240,160,192,0,128,622,864,880,768,640,384,448,0,256,2120,
%U A096794 2932,2976,2624,2080,1632,896,1024,0,512,7338,10112,10248,9024,7280
%N A096794 Triangle read by rows: a(n,k) = number of Dyck n-paths such that number of DUs at level 1 plus number of UDs at level 2 is k, 0<=k<=n-1.
%C A096794 Column k has g.f. F(x)^(k+1)*(2y)^k where F(x)=(1-sqrt(1-4*x))/(3-sqrt(1-4*x)) is the g.f. for Fine's sequence A000957.
%F A096794 G.f. (1 - (1 - 4*x)^(1/2))/(3 - 2y + (2y-1)(1 - 4*x)^(1/2) ) = Sum_{n>=1, k>=0} a(n, k) x^n y^k.
%e A096794 Table begins
%e A096794 \ k 0, 1, 2, ...
%e A096794 n
%e A096794 1 | 1
%e A096794 2 | 0, 2
%e A096794 3 | 1, 0, 4
%e A096794 4 | 2, 4, 0, 8
%e A096794 5 | 6, 8, 12, 0, 16
%e A096794 6 | 18, 26, 24, 32, 0, 32
%e A096794 7 | 57, 80, 84, 64, 80, 0, 64
%e A096794 a(4,1) = 4 because UudUUDDD, UUUDDudD, UduUUDDD, UUUDDduD each contain one
%e A096794 relevant turn (in small type).
%Y A096794 Row sums are the Catalan numbers A000108.
%Y A096794 Adjacent sequences: A096791 A096792 A096793 this_sequence A096795 A096796 A096797
%Y A096794 Sequence in context: A062173 A004558 A002349 this_sequence A106375 A131667 A086802
%K A096794 nonn,tabl
%O A096794 1,3
%A A096794 David Callan (callan(AT)stat.wisc.edu), Aug 17 2004
%I A106375
%S A106375 2,1,0,4,2,4,4,1,0,0,8,4,8,24,18,36,48,40,36,24,8,1,0,0,0,16,8,16,48,
%T A106375 100,136,240,528,616,1152,1936,2466,3716,4912,5840,7088,7768,7696,7120,
%U A106375 5796,4056,2464,1232,456,112,16,1,0,0,0,0,32,16,32,96,200,528,736,1632
%N A106375 Triangle read by rows: T(n,k) is the number of binary trees (each vertex has 0, or 1 left, or 1 right, or 2 children) with k edges and all leaves at level n.
%C A106375 Row n has 2^(n+1)-2 terms. In row n first nonzero term is T(n,n)=2^n and last nonzero term is T(n,2^(n+1)-2)=1. Row sums yield A051179. Column sums yield A106376.
%F A106375 T(n, k)=2T(n-1, k-1) + sum(T(n-1, j)T(n-1, k-2-j), j=1..k-3) (n, k>=2); T(1, 1)=2, T(1, 2)=1, T(1, k)=0 for k>=3, T(n, 1)=0 for n>=2. Generating polynomial P[n](t) of row n is given by rec. eq. P[n]=2tP[n-1]+(t*P[n-1])^2, P[0]=1.
%e A106375 T(3,3)=8 because we have eight paths of length 3 (each edge can have two orientations).
%e A106375 Triangle begins:
%e A106375 2,1;
%e A106375 0,4,2,4,4,1;
%e A106375 0,0,8,4,8,24,18,36,48,40,36,24,8,1;
%p A106375 P[0]:=1: for n from 1 to 5 do P[n]:=sort(expand(2*t*P[n-1]+t^2*P[n-1]^2)) od: for n from 1 to 5 do seq(coeff(P[n],t^k),k=1..2^(n+1)-2) od; # yields sequence in triangular form
%Y A106375 Cf. A051179, A106376.
%Y A106375 Adjacent sequences: A106372 A106373 A106374 this_sequence A106376 A106377 A106378
%Y A106375 Sequence in context: A004558 A002349 A096794 this_sequence A131667 A086802 A092488
%K A106375 nonn,tabf
%O A106375 1,1
%A A106375 Emeric Deutsch (deutsch(AT)duke.poly.edu), May 05 2005
%I A131667
%S A131667 2,1,0,4,3,0,6,5,0,8,7,0,10,9,0,12,11,0,14,13,0,16,15,0,18,17,0,20,19,0,
%T A131667 22,21,0,24,23,0,26,25,0,28,27,0,30,29,0,32,31,0,34,33,0,36,35,0,38,37,
%U A131667 0,40,39,0,42,41,0,44,43,0,46,45,0,48,47,0,50,49,0
%V A131667 2,-1,0,4,-3,0,6,-5,0,8,-7,0,10,-9,0,12,-11,0,14,-13,0,16,-15,0,18,-17,0,20,-19,0,22,
%W A131667 -21,0,24,-23,0,26,-25,0,28,-27,0,30,-29,0,32,-31,0,34,-33,0,36,-35,0,38,-37,0,40,-39,
%X A131667 0,42,-41,0,44,-43,0,46,-45,0,48,-47,0,50,-49,0
%N A131667 List of triples 2n, 1-2n, 0, n >= 1.
%Y A131667 Adjacent sequences: A131664 A131665 A131666 this_sequence A131668 A131669 A131670
%Y A131667 Sequence in context: A002349 A096794 A106375 this_sequence A086802 A092488 A068527
%K A131667 sign
%O A131667 1,1
%A A13