%I A001787 M3444 N1398
%S A001787 0,1,4,12,32,80,192,448,1024,2304,5120,11264,24576,53248,114688,245760,
%T A001787 524288,1114112,2359296,4980736,10485760,22020096,46137344,96468992,
%U A001787 201326592,419430400,872415232,1811939328,3758096384,7784628224
%N A001787 n*2^(n-1).
%C A001787 Number of edges in n-dimensional hypercube.
%C A001787 Number of 132-avoiding permutations of [n+2] containing exactly one 123
pattern. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 13 2001
%C A001787 Number of ways to place n-1 nonattacking kings on a 2 X 2(n-1) chessboard
for n >= 2 - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com),
May 22 2001
%C A001787 Arithmetic derivative of 2^n: a(n) = A003415(A000079(n)). - Reinhard
Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 26 2002
%C A001787 (-1) times determinant of matrix A_{i,j} = -|i-j|, 0<=i,j<=n.
%C A001787 a(n)= number of ones in binary numbers 1 to 111...1 (n bits). a(n) =
A000337(n)-A000337(n-1) for n = 2,3,... - Emeric Deutsch (deutsch(AT)duke.poly.edu),
May 24 2003
%C A001787 The number of 2 X n 0-1 matrices containing n+1 1's and having no zero
row or column. The number of spanning trees of the complete bipartite
graph K(2,n). This is the case m = 2 of K(m,n). See A072590. - W.
Edwin Clark (eclark(AT)math.usf.edu), May 27 2003
%C A001787 Binomial transform of [0,1,2,3,4,5,...]. Without the initial 0, binomial
transform of odd numbers.
%C A001787 With an additional leading zero, [0,0,1,4,...] this is the binomial transform
of the integers repeated A004526. Its formula is then (2^n(n-1)+0^n)/
4. - Paul Barry (pbarry(AT)wit.ie), May 20 2003
%C A001787 PSumSIGN transform of A053220. PSumSIGN transform is A045883. Binomial
transform is A027471(n+1). - Michael Somos, Jul 10 2003
%C A001787 Number of zeros in all different (n+1)-bit integers. - Ralf Stephan (ralf(AT)ark.in-berlin.de),
Aug 02 2003
%C A001787 Final element of a summation table (as opposed to a difference table)
whose first row consists of integers 0 through n(or first n+1 nonnegative
integers A001477);Illustrating the case n=5:
%C A001787 0...1...2...3...4...5
%C A001787 ..1...3...5...7...9
%C A001787 ....4...8...12..16
%C A001787 ......12..20..28
%C A001787 ........32..48
%C A001787 ..........80 and final element is a(5)=80. - Lekraj Beedassy (blekraj(AT)yahoo.com),
Jun 03 2004
%C A001787 This sequence and A001871 arise in counting ordered trees of height at
most k where only the right-most branch at the root actually achieves
this height and the count is by the number of edges, with k = 3 for
this sequence and k = 4 for A001871.
%C A001787 Let R be a binary relation on the power set P(A) of a set A having n
= |A| elements such that for all elements x,y of P(A), xRy if x is
a proper subset of y and there are no z in P(A) such that x is a
proper subset of z and z is a proper subset of y. Then a(n) = |R|.
- Ross La Haye (rlahaye(AT)new.rr.com), Sep 21 2004
%C A001787 Number of 2 X n binary matrices avoiding simultaneously the right angled
numbered polyomino patterns (ranpp) (00;1) and (10;1). An occurrence
of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,
j2), a(i2,j1)) where i1<i2, j1<j2 and these elements are in same
relative order as those in the triple (x,y,z). - Sergey Kitaev (kitaev(AT)ms.uky.edu),
Nov 11 2004
%C A001787 Number of subsequences 00 in all binary words of length n+1. Example:
a(2)=4 because in 000,001,010,011,100,101,110,111 the sequence 00
occurs 4 times. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr
04 2005
%C A001787 Let M=[1,i;i,1], i=sqrt(-1). Then g.f.=x/det(I-xM). - Paul Barry (pbarry(AT)wit.ie),
Apr 27 2005
%C A001787 If you expand the n-factor expression (a+1)(b+1)(c+1)...(z+1), there
are a(n) variables in the result. For example, the 3-factor expression
(a+1)(b+1)(c+1) expands to abc+ab+ac+bc+a+b+c+1 with a(3) = 12 variables.
- David W. Wilson (davidwwilson(AT)comcast.net), May 08 2005
%C A001787 An inverse Chebyshev transform of n^2, where g(x)->(1/sqrt(1-4x^2))g(xc(x^2)),
c(x) the g.f. of A000108. - Paul Barry (pbarry(AT)wit.ie), May 13
2005
%C A001787 Sequences A018215 and A058962 interleaved. - Graeme McRae (g_m(AT)mcraefamily.com),
Jul 12 2006
%C A001787 The number of never-decreasing positive integer sequences of length n
with a maximum value of 2*n. - Ben Thurston (benthurston27(AT)yahoo.com),
Nov 13 2006
%C A001787 Total size of all the subsets of an n-element set. For example, a 2-element
set has 1 subset of size 0, 2 subsets of size 1 and 1 of size 2.
- Ross La Haye (rlahaye(AT)new.rr.com), Dec 30 2006
%C A001787 Convolution of the natural numbers [A000027] and A045623 beginning [0,
1,2,5...]. - Ross La Haye (rlahaye(AT)new.rr.com), Feb 03 2007
%C A001787 If M is the matrix (given by rows) [2,-1;0,2] then the sequence gives
the (1,2) entry in M^n. - Antonio M. Oller (oller(AT)unizar.es),
May 21 2007
%C A001787 If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for
n>0, a(n) is equal to the number of (n+1)-subsets of X intersecting
each X_i (i=1,2,...,n). - Milan R. Janjic (agnus(AT)blic.net), Jul
21 2007
%C A001787 Number of n-permutations of 3 objects u,v,w, with repetition allowed,
containing exactly one u. Example: a(2)=4 because we have uv, vu,
uw and wu. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 27
2007
%C A001787 A member of the family of sequences defined by a(n) = n*[c(1)*...c(r)]^(n-1);
c(i) integer. This sequence has c(1)=2, A027471 has c(1)=3. - Ctibor
O. ZIZKA (ctibor.zizka(AT)seznam.cz), Feb 23 2008
%C A001787 Sum(n>0,1/a(n))=2log(2) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com),
Feb 10 2009]
%C A001787 Equals the Jacobsthal sequence A001045 convolved with A003945: (1, 3,
6, 12,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23
2009]
%C A001787 Starting with offset 1 = A059570: (1, 2, 6, 14, 34,...) convolved with
(1, 2, 2, 2,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May
23 2009]
%D A001787 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001787 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001787 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,
National Bureau of Standards Applied Math. Series 55, 1964 (and various
reprintings), p. 796.
%D A001787 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of
combinatorial proof, M.A.A. 2003, id. 131.
%D A001787 M. Elkadi and B. Mourrain, Symbolic-numeric methods for solving polynomial
equations and applications, Chap 3. of A. Dickenstein and I. Z. Emiris,
eds., Solving Polynomial Equations, Springer, 2005, pp. 126-168.
See p. 152.
%D A001787 F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424.
%D A001787 A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib.
Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=4, q=-4.
%D A001787 Jones, C. W.; Miller, J. C. P.; Conn, J. F. C.; Pankhurst, R. C.; Tables
of Chebyshev polynomials. Proc. Roy. Soc. Edinburgh. Sect. A. 62,
(1946). 187-203.
%D A001787 T. Y. Lam, On the diagonalization of quadratic forms, Math. Mag., 72
(1999), 231-235 (see page 234).
%D A001787 W. Lang, On polynomials related to powers of the generating function
of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (38)
and (45), lhs, m=4.
%D A001787 Ross La Haye, Binary Relations on the Power Set of an n-Element Set,
Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From
Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]
%H A001787 Franklin T. Adams-Watters, <a href="b001787.txt">Table of n, a(n) for
n = 0..500</a>
%H A001787 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A001787 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative
Functions</a>
%H A001787 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National
Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
[alternative scanned copy].
%H A001787 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A001787 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001787 D. W. Bass and I. H. Sudborough, <a href="http://jgaa.info/">Hamilton
decompositions and (n/2)-factorizations of hypercubes</a>, J Graph
Algor. Appl. 7(2003) 79-98.
%H A001787 D. Callan, <a href="http://arXiv.org/abs/math.CO/0211380">A recursive
bijective approach to counting permutations...</a>
%H A001787 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
Sequences realized by oligomorphic permutation groups</a>, J. Integ.
Seqs. Vol. 3 (2000), #00.1.5.
%H A001787 F. Ellermann, <a href="a001792.txt">Illustration of binomial transforms</
a>
%H A001787 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=408">
Encyclopedia of Combinatorial Structures 408</a>
%H A001787 S. Kitaev, <a href="http://www.integers-ejcnt.org/vol4.html">On multi-avoidance
of right angled numbered polyomino patterns</a>, Integers: Electronic
Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
%H A001787 S. Kitaev, <a href="http://www.ms.uky.edu/%7Emath/MAreport/4-ser.ps">
On multi-avoidance of right angled numbered polyomino patterns</a>
, University of Kentucky Research Reports (2004).
%H A001787 M. L. Perez et al., eds., <a href="http://www.gallup.unm.edu/~smarandache/
">Smarandache Notions Journal</a>
%H A001787 A. Robertson, <a href="http://www.dmtcs.org/volumes/abstracts/dm030402.abs.html">
Permutations containing and avoiding 123 and 132 patterns</a>
%H A001787 A. Robertson, H. S. Wilf and D. Zeilberger, Permutation patterns and
continued fractions, <a href="http://www.combinatorics.org/">Electr.
J. Combin.</a> 6, 1999, #R38.
%H A001787 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
Hypercube.html">Hypercube</a>
%H A001787 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
LeibnizHarmonicTriangle.html">Leibniz Harmonic Triangle</a>
%H A001787 Thomas Wieder, The number of certain k-combinations of an n-set, <a href="http:/
/www.math.nthu.edu.tw/~amen/">Applied Mathematics Electronic Notes</
a>, vol. 8 (2008).
%H A001787 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to
Chebyshev polynomials.</a>
%F A001787 a(n) = sum(k=1, n, k*binomial(n, k)). - Benoit Cloitre (benoit7848c(AT)orange.fr),
Dec 06 2002
%F A001787 E.g.f. xexp(2x) - Paul Barry (pbarry(AT)wit.ie), Apr 10 2003
%F A001787 G.f.: x/(1-2x)^2. a(n)=2a(n-1)+2^(n-1). a(2n)= n4^n, a(2n+1)= (2n+1)4^n.
%F A001787 Starting 1, 1, 4, 12, .. this is 0^n+n2^(n-1), the binomial transform
of the 'pair-reversed' natural numbers A004442 - Paul Barry (pbarry(AT)wit.ie),
Jul 24 2003
%F A001787 Convolution of [1, 2, 4, 8, ...] with itself. - Jon Perry (perry(AT)globalnet.co.uk),
Aug 07 2003
%F A001787 The signed version of this sequence, n(-2)^(n-1), is the inverse binomial
transform of n(-1)^(n-1) (alternating sign natural numbers). - Paul
Barry (pbarry(AT)wit.ie), Aug 20 2003
%F A001787 a(n-1)=sum{k=0..n, 2^(n-k-1)C(n-k, k)C(1, (k+1)/2)(1-(-1)^k)/2}-0^n/4.
- Paul Barry (pbarry(AT)wit.ie), Oct 15 2004
%F A001787 a(n)=sum{k=0..floor(n/2), binomial(n, k)(n-2k)^2}; - Paul Barry (pbarry(AT)wit.ie),
May 13 2005
%F A001787 a(n+2) = A049611(n+2) - A001788(n). Floretion Algebra Multiplication
Program, FAMP Code: 1vessum(pos)seq[A], 1vessum(neg)seq[A] and 1vessumseq[A]
(= (a(n)) from 2nd term) with A = + .5'i + .5i' + .5'ij' + .5'ki'
+ 2e. Sumtype is set to: default (ver. f) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de),
Aug 02 2005
%F A001787 a(n)=n!sum{k=0..n, 1/((k - 1)!(n - k)!)} - Paul Barry (pbarry(AT)wit.ie),
Mar 26 2003
%F A001787 a(n) = sum(binomial(n+1,j)*(n+1-j),j=0..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Aug 22 2006
%F A001787 a(n+1)=Sum_{k, 0<=k<=n}4^k*A109466(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Nov 13 2006
%F A001787 Row sums of A130300 starting (1, 4, 12, 32,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com),
May 20 2007
%F A001787 Equals row sums of triangle A134083. Equals A002064(n) + (2^n - 1). -
Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 07 2007
%F A001787 a(n)=4*a(n-1)-4*a(n-2), a(0)=0, a(1)=1. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Nov 16 2008]
%F A001787 a(n) is the number of ways to split {1,2,...n-1} into two (possibly empty)
complimentary intervals {1,2,...i} and {i+1,i+2,...n-1} and then
select a subset from each interval. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org),
Jan 31 2009]
%e A001787 a(2)=4 since 2314, 2341,3124 and 4123 are the only 132-avoiding permutations
of 1234 containing exactly one increasing subsequence of length 3.
%p A001787 spec := [S, {B=Set(Z, 0 <= card), S=Prod(Z, B, B)}, labeled]: seq(combstruct[count](spec,
size=n), n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Oct 09 2006
%p A001787 a:=n->sum (2^(n-1),j=1..n): seq(a(n),n=0..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 27 2007
%p A001787 A001787:=1/(2*z-1)^2; [S. Plouffe in his 1992 dissertation, dropping
the initial zero.]
%p A001787 with (combinat): c := n -> stirling2(n,2): b := n -> if n<2 then 1; else
c(n)-c(n-1); fi: a := n -> add(b(i)*c(n-i), i=1..n-1): seq(a(n),n=2..31);
- Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 10 2007
%p A001787 with(finance):seq(add(futurevalue( 1, 1, n),k=0..n),n=- 1..28); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008
%p A001787 with(finance):seq(add(futurevalue( 2, 1, n),k=0..n)/2,n=-1..28); - Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Jun 20 2008
%t A001787 Table[Sum[Binomial[n, i] i, {i, 0, n}], {n, 0, 30}] [From Geoffrey Critzer
(critzer.geoffrey(AT)usd443.org), Mar 18 2009]
%o A001787 (PARI) a(n)=if(n<0,0,n*2^(n-1))
%o A001787 (Other) sage: [lucas_number1(n,4,4) for n in xrange(0, 30)]# [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%Y A001787 Partial sums of A001792. Cf. A053109, A001788, A001789. A058922(n+1)
= 4*A001787(n).
%Y A001787 Row sums of triangle in A003506. Equals A090802(n, 1).
%Y A001787 Cf. A000337, A130300, A134083, A002064.
%Y A001787 Three other versions, essentially identical, are A085750, A097067, A118442.
%Y A001787 Cf. A027471.
%Y A001787 Cf. A003945, A059670.
%Y A001787 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12
2009: (Start)
%Y A001787 Equals the first left hand column of A167591.
%Y A001787 (End)
%Y A001787 Sequence in context: A085750 A097067 A139756 this_sequence A118442 A038592
A048776
%Y A001787 Adjacent sequences: A001784 A001785 A001786 this_sequence A001788 A001789
A001790
%K A001787 nonn,easy,nice
%O A001787 0,3
%A A001787 N. J. A. Sloane (njas(AT)research.att.com).
%E A001787 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Mar 11 2009
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