Search: id:A000295 Results 1-1 of 1 results found. %I A000295 M3416 N1382 %S A000295 0,0,1,4,11,26,57,120,247,502,1013,2036,4083,8178,16369,32752, %T A000295 65519,131054,262125,524268,1048555,2097130,4194281,8388584, %U A000295 16777191,33554406,67108837,134217700,268435427,536870882 %N A000295 Eulerian numbers 2^n - n - 1. (Column 2 of Euler's triangle A008292.) %C A000295 Number of Dyck paths of semilength n having exactly one long ascent (i.e. ascent of length at least two). Example: a(4)=11 because among the 14 Dyck paths of semilength 4, the paths that do not have exactly one long ascent are UDUDUDUD (no long ascent), UUDDUUDD and UUDUUDDD (two long ascents). Here U=(1,1) and D=(1,-1). Also number of ordered trees with n edges having exactly one branch node (i.e. vertex of outdegree at least two). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2004 %C A000295 Number of permutations of {1,2,...,n} with exactly one descent (i.e. permutations (p(1),p(2),...,p(n)) such that #{i: p(i)>p(i+1)}=1). E.g. a(3)=4 because the permutations of {1,2,3} with one descent are 132, 213, 231 and 312. %C A000295 A107907(a(n+2)) = A000079(n+2). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 28 2005 %C A000295 a(n+1) is the convolution of nonnegative integers (A001477) and powers of two (A000079) - Graeme McRae (g_m(AT)mcraefamily.com), Jun 07 2006 %C A000295 Number of partitions of an n-set having exactly one block of size > 1. Example: a(4)=11 because, if the partitioned set is {1,2,3,4}, then we have 1234, 123|4, 124|3, 134|2, 1|234, 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 28 2006 %C A000295 n divides a(n+1) for n = A014741(n) = {1, 2, 6, 18, 42, 54, 126, 162, 294, 342, 378, 486, 882, 1026, ...}. - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 03 2006 %C A000295 (Number of permutations avoiding patterns 321, 2413, 3412, 21534) minus one. - J.-L. Baril (barjl(AT)u-bourgogne.fr), Nov 01 2007, Mar 21 2008 %C A000295 The chromatic invariant of the prism graph P_n for n=>3. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 29 2008] %C A000295 look ma, no need for power! just plus and minus, simplify! [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 10 2008] %C A000295 a(n) is the number of binary sequences of length n having at least two 0's. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Feb 11 2009] %C A000295 Decimal integer corresponding to the result of XORing the binary representation of 2^n - 1 and the binary representation of n with leading zeros. This sequence and a few others are syntactically similar. For n > 0, let D(n) denote the decimal integer corresponding to the binary number having n consecutive 1's. Then D(n).OP.n represents the n-th term of a sequence when .OP. stands for a binary operator such as '+', '-', '*', 'quotentof', 'mod', 'choose'. We then get the various sequences A136556, A082495, A082482, A066524, A000295, A052944. Another syntactically similar sequence results when we take the n-th term as f(D(n)).OP.f(n). For example if f='factorial' and .OP.='/', we get (A136556)(A000295) ; if f='squaring' and .OP.='-', we get (A000295)(A052944). [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Mar 30 2009] %C A000295 Chromatic invariant of the prism graph Y_n %C A000295 Starting with 1 = A000975 convolved with [1, 2, 2, 2,...] [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 02 2009] %C A000295 Contribution from Michael Vielhaber (vielhaber(AT)gmail.com), Nov 18 2009: (Start) %C A000295 Number of labellings of a full binary tree of height n-1, such that %C A000295 each path from root to any leaf contains each label from {1,2,..,n-1} exactly once. (End) %D A000295 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A000295 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000295 O. Bottema, Problem #562, Nieuw Archief voor Wiskunde, 28 (1980) 115. %D A000295 F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151. %D A000295 Pascal Floquet, Serge Domenech and Luc Pibouleau, "Combinatorics of Sharp Separation System synthesis : Generating functions and Search Efficiency Criterion", Industrial Engineering and Chemistry Research, 33, pp. 440-443, 1994 %D A000295 Pascal Floquet, Serge Domenech, Luc Pibouleau and Said Aly, "Some Complements in Combinatorics of Sharp Separation System Synthesis", American Institute of Chemical Engineering Journal, 39(6), pp. 975-978, 1993. %D A000295 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990. %D A000295 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 34. %D A000295 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215. %D A000295 D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 20 (1968), 8-16. %H A000295 T. D. Noe, Table of n, a(n) for n=0..500 %H A000295 Index entries for sequences related to linear recurrences with constant coefficients %H A000295 P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5. %H A000295 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 388 %H A000295 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000295 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A000295 Eric W. Weisstein, Chromatic Invariant. %H A000295 Eric Weisstein's World of Mathematics, Chromatic Invariant %F A000295 G.f.: x^2/((1-2*x)*(1-x)^2). a(1)=0, a(n)=2*a(n-1)+n-1. %F A000295 a(n)=sum{k=2..n, binomial(n, k) } - Paul Barry (pbarry(AT)wit.ie), Jun 05 2003 %F A000295 a(n+1)=sum(1<=i<=n, sum(1<=j<=i, C(i, j))) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 07 2003 %F A000295 a(n+1)=2^n*sum(k=0, n, k/2^k) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 26 2003 %F A000295 a(0)=0, a(1)=0, a(n) = Sum i=0..n-1 i+a(i) for i > 1 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 12 2004 %F A000295 a(n+1)=sum{k=0..n, (n-k)2^k}=sum{k=0..n, k*2^(n-k)} - Paul Barry (pbarry(AT)wit.ie), Jul 29 2004 %F A000295 a(n)=sum{k=0..n, binomial(n, k+2)}; a(n+2)=sum{k=0..n, binomial(n+2, k+2)}. - Paul Barry (pbarry(AT)wit.ie), Aug 23 2004 %F A000295 a(n)=sum{k=0..floor((n-1)/2), binomial(n-k-1, k+1)2^(n-k-2)*(-1/2)^k} - Paul Barry (pbarry(AT)wit.ie), Oct 25 2004 %F A000295 a(0)=0, a(1)=0, a(n)=3*a(n-1)-2*a(n-2)+1 - Miklos Kristof (kristmikl(AT)freemail.hu), Mar 09 2005 %F A000295 G.f.=exp(x)[exp(x)-1-x]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 28 2006 %F A000295 a(0) = 0; a(n)=stirling2(n,2)+a(n-1). - Thomas Wieder (thomas.wieder(AT)t-online.de), Feb 18 2007 %F A000295 Equals row sums of triangle A130128 starting (1, 4, 11, 26, 57,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 11 2007 %F A000295 Starting (1, 4, 11, 26,...) gives row sums of triangle A130330 which is composed of (1,3,7,15...) in every column, thus: row sums of (1; 3,1; 7,3,1;...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 24 2007 %F A000295 a(n) = 2*a(n-1)+n-1. - Graeme McRae (g_m(AT)mcraefamily.com), Jun 15 2007 %F A000295 Row sums of triangle A131768 starting (1, 4, 11, 26, 57, 120,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 13 2007 %F A000295 Sequence starting (1, 4, 11, 26, 57,...) = A130321 * (1, 2, 3,...). Sequence starting (1, 4, 11, 26, 57,...) = binomial transform of (1, 3, 4, 4, 4,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 27 2007 %F A000295 Row sums of triangle A131816 starting (1, 4, 11, 26,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 30 2007 %F A000295 a(n) = A000325(n)-1. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 29 2008] %F A000295 a(0) = 0, a(n) = sum{k=0..n-1, 2^k - 1} [From Doug Bell (bell.doug(AT)gmail.com), Jan 19 2009] %F A000295 a(n) =A000225-n . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2009] %p A000295 [ seq(2^n-n-1,n=1..50) ]; %p A000295 a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=3*a[n-1]-2*a[n-2]+1 od: seq(a[n], n=0..50); (Kristof) %p A000295 seq(add(binomial(n, k)*(bell(k-n)), k=2..n), n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 01 2006 %p A000295 a:=n->sum (2^j-1,j=1..n): seq(a(n),n=-1..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 27 2007 %p A000295 with(combinat): a:=n->(sum((stirling2(j,2)),j=0..n)): seq(a(n),n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007 %p A000295 A000295:=-z/(2*z-1)/(z-1)**2; [S. Plouffe in his 1992 dissertation.] %p A000295 a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=n+2*a[n-1]-1 od: seq(a[n], n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008 %t A000295 lst={0, 0};s=0;Do[s+=2^n-1;AppendTo[lst, s], {n, 5!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 30 2008] %t A000295 ...and/or... lst={};s=0;Do[s+=(s-n);AppendTo[lst, s], {n, 5!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 10 2008] %t A000295 Table[Sum[ Binomial[n + 2, k + 2], {k, 0, n}], {n, -2, 27}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009] %t A000295 s = 0; lst = {s}; Do[s += StirlingS2[n, 2]; AppendTo[lst, s], {n, 1, 29, 1}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 14 2009] %o A000295 (Other) sage: [gaussian_binomial(n,1,2)-n for n in xrange(0,30)] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 29 2009] %Y A000295 Cf. A008949, A000079, A000225, A002663, A002664, A035039-A035042, A008292 (the main entry for the Eulerian numbers). %Y A000295 Cf. A000108, A014741, A130128, A130330, A131768, A130321, A131816. %Y A000295 Partial sums of A000225. %Y A000295 Row sums of triangle A014473. Second column of triangles A112493 and A112500. %Y A000295 Cf. A000325. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 29 2008] %Y A000295 A000295-A002662=A000217 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Feb 11 2009] %Y A000295 Row sums of A143291. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jun 01 2009] %Y A000295 Cf. A000975. %Y A000295 Sequence in context: A027660 A002940 A030196 this_sequence A125128 A130103 A034334 %Y A000295 Adjacent sequences: A000292 A000293 A000294 this_sequence A000296 A000297 A000298 %K A000295 nonn,easy,nice,new %O A000295 0,4 %A A000295 N. J. A. Sloane (njas(AT)research.att.com). Search completed in 0.003 seconds