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A001523
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Number of stacks, or planar partitions of n; also weakly unimodal partitions of n.
(Formerly M1102 N0420)
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18
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) counts stacks of integer-length boards of total length n where no board overhangs the board underneath.
A006330(n)+a(n)=A000712(n). - Michael Somos, Jul 22 2003
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 (perry(AT)globalnet.co.uk), Jul 25 2003
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REFERENCES
| F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs. Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
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).
A. D. Sokal, The leading root of the partial theta function, Arxiv preprint arXiv:1106.1003, 2011.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see section 2.5 on page 76.
E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
H. Bottomley, Illustration of initial terms
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 46
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FORMULA
| a(n) = Sum(1 <= k <= n, f(k, n-k)), where f(n, k) (=A054250) = 1 if k = 0; Sum(1 <= j <= min(n, k); (n-j+1) f(j, k-j)) if k > 0.
a(n)=sum_k[A059623(n, k)] for n>0 - Henry Bottomley (se16(AT)btinternet.com), Feb 01 2001
G.f.: (Sum_{k>0} -(-1)^k x^(k(k+1)/2))/(Product_{k>0} (1-x^k))^2.
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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
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MATHEMATICA
| max = 38; f[x_] := Sum[ -(-1)^k x^(k (k + 1)/2), {k, 1, max}] / Product[ 1 - x^k, {k, 1, max}]^2; se = Series[ f[x], {x, 0, max}]; a[n_] := SeriesCoefficient[se, n]; a[0] = 1; Table[ a[n], {n, 0, max}] (* From Jean-François Alcover, Jan 25 2012, after g.f. *)
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PROG
| (PARI) a(n)=if(n<1, 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))
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CROSSREFS
| Cf. A054250, A059618, A059623, A001522, A001524.
Cf. A000569. Bisections give A100505, A100506.
Sequence in context: A191630 A125513 A054174 * A000126 A182716 A143281
Adjacent sequences: A001520 A001521 A001522 * A001524 A001525 A001526
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Formula and more terms from David W. Wilson (davidwwilson(AT)comcast.net) May 05 2000.
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