This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A001523 Number of stacks, or planar partitions of n; also weakly unimodal partitions of n. (Formerly M1102 N0420) 28
 1, 1, 2, 4, 8, 15, 27, 47, 79, 130, 209, 330, 512, 784, 1183, 1765, 2604, 3804, 5504, 7898, 11240, 15880, 22277, 31048, 43003, 59220, 81098, 110484, 149769, 202070, 271404, 362974, 483439, 641368, 847681, 1116325, 1464999, 1916184, 2498258 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) counts stacks of integer-length boards of total length n where no board overhangs the board underneath. Number of graphical partitions on 2n nodes that contain a 1. E.g. a(3)=4 and so there are 4 graphical partitions of 6 that contain a 1, namely (111111), (21111), (2211) and (3111). Only (222) fails. - Jon Perry, Jul 25 2003 It would seem from Stanley that he regards a(0)=0 for this sequence and A001522. - Michael Somos, Feb 22 2015 In the article by Auluck is a typo in the formula (24), 2*Pi is missing in an exponent on the left side of the equation for Q(n). The correct term is exp(2*Pi*sqrt(n/3)), not just exp(sqrt(n/3)). - Vaclav Kotesovec, Jun 22 2015 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see section 2.5 on page 76. LINKS T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe) F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686, g(x). F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs (annotated scanned copy) Shouvik Datta, M. R. Gaberdiel, W. Li, C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1. H. Bottomley, Illustration of initial terms P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 46 R. C. Rhoades, Strongly Unimodal Sequences and Mixed Mock Modular Forms A. D. Sokal, The leading root of the partial theta function, arXiv preprint arXiv:1106.1003 [math.CO], 2011. E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158. FORMULA a(n) = Sum_{k=1..n} f(k, n-k), where f(n, k) (= A054250) = 1 if k = 0; Sum_{j=1..min(n, k)} (n-j+1) f(j, k-j)) if k > 0. - David W. Wilson, May 05 2000 a(n) = Sum_{k} A059623(n, k) for n > 0. - Henry Bottomley, Feb 01 2001 A006330(n) + a(n) = A000712(n). - Michael Somos, Jul 22 2003 G.f.: 1 + (Sum_{k>0} -(-1)^k x^(k(k+1)/2))/(Product_{k>0} (1-x^k))^2. - Michael Somos, Jul 22 2003 G.f.: 1 + Sum_{n>=1} (x^n / ( ( Product_{k=1..n-1} (1 - x^k)^2 ) * (1-x^n) ) ). - Joerg Arndt, Oct 01 2012 a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)) [Auluck, 1951]. - Vaclav Kotesovec, Jun 22 2015 EXAMPLE For a(4)=8 we have the following stacks: x x x. .x x x. .x x.. .x. ..x xx x xx xx xxx xxx xxx xx xxxx G.f. = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 15*x^5 + 27*x^6 + 47*x^7 + 79*x^8 + ... MAPLE b:= proc(n, i) option remember;       `if`(i>n, 0, `if`(irem(n, i)=0, 1, 0)+       add(b(n-i*j, i+1)*(j+1), j=0..n/i))     end: a:= n-> `if`(n=0, 1, b(n, 1)): seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 2014 MATHEMATICA max = 40; s = 1 + Sum[(-1)^(k + 1)*q^(k*(k + 1)/2), {k, 1, max}] / QPochhammer[q]^2 + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Jan 25 2012, updated Nov 29 2015 *) b[n_, i_] := b[n, i] = If[i>n, 0, If[Mod[n, i]==0, 1, 0] + Sum[b[n-i*j, i+1]*(j+1), {j, 0, n/i}]]; a[n_] := If[n==0, 1, b[n, 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 24 2015, after Alois P. Heinz *) PROG (PARI) {a(n) = if( n<1, n==0, polcoeff( sum(k=1, (sqrt(1 + 8*n) - 1)\2, -(-1)^k * x^((k + k^2)/2)) / eta(x + x * O(x^n))^2 , n))}; /* Michael Somos, Jul 22 2003 */ (Python) def b(n, i):     if i>n: return 0     if n%i==0: x=1     else: x=0     return x + sum([b(n - i*j, i + 1)*(j + 1) for j in xrange(n/i + 1)]) def a(n): return 1 if n==0 else b(n, 1) # Indranil Ghosh, Jun 09 2017, after Maple code by Alois P. Heinz CROSSREFS Cf. A054250, A059618, A059623, A001522, A001524. Cf. A000569. Bisections give A100505, A100506. Row sums of A247255. Sequence in context: A054174 A239890 A222038 * A222039 A222148 A222040 Adjacent sequences:  A001520 A001521 A001522 * A001524 A001525 A001526 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS More terms from David W. Wilson, May 05 2000 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.