Search: id:A001523
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%I A001523 M1102 N0420
%S A001523 1,1,2,4,8,15,27,47,79,130,209,330,512,784,1183,1765,2604,3804,5504,
%T A001523 7898,11240,15880,22277,31048,43003,59220,81098,110484,149769,202070,
%U A001523 271404,362974,483439,641368,847681,1116325,1464999,1916184,2498258
%N A001523 Number of stacks, or planar partitions of n; also weakly unimodal partitions 
               of n.
%C A001523 a(n) counts stacks of integer-length boards of total length n where no 
               board overhangs the board underneath.
%C A001523 A006330(n)+a(n)=A000712(n). - Michael Somos, Jul 22 2003
%C A001523 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
%D A001523 F. C. Auluck, On some new types of partitions associated with generalized 
               Ferrers graphs. Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
%D A001523 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001523 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001523 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see 
               section 2.5 on page 76.
%D A001523 E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.
%H A001523 T. D. Noe, <a href="b001523.txt">Table of n, a(n) for n=0..1000</a>
%H A001523 H. Bottomley, <a href="a1523.gif">Illustration of initial terms</a>
%H A001523 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               46
%F A001523 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.
%F A001523 a(n)=sum_k[A059623(n, k)] for n>0 - Henry Bottomley (se16(AT)btinternet.com), 
               Feb 01 2001
%F A001523 G.f.: (Sum_{k>0} -(-1)^k x^(k(k+1)/2))/(Product_{k>0} (1-x^k))^2.
%e A001523 For a(4)=8 we have the following stacks:
%e A001523 x
%e A001523 x x. .x
%e A001523 x x. .x x.. .x. ..x xx
%e A001523 x xx xx xxx xxx xxx xx xxxx
%o A001523 (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))
%Y A001523 Cf. A054250, A059618, A059623, A001522, A001524.
%Y A001523 Cf. A000569. Bisections give A100505, A100506.
%Y A001523 Sequence in context: A003241 A125513 A054174 this_sequence A000126 A143281 
               A098057
%Y A001523 Adjacent sequences: A001520 A001521 A001522 this_sequence A001524 A001525 
               A001526
%K A001523 nonn,nice,easy
%O A001523 0,3
%A A001523 N. J. A. Sloane (njas(AT)research.att.com).
%E A001523 Formula and more terms from David W. Wilson (davidwwilson(AT)comcast.net) 
               May 05 2000.

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