Search: id:A001522
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%I A001522 M0644 N0238
%S A001522 0,1,1,1,2,3,5,7,10,14,19,26,35,47,62,82,107,139,179,230,293,372,470,
%T A001522 591,740,924,1148,1422,1756,2161,2651,3244,3957,4815,5844,7075,8545,
%U A001522 10299,12383,14859,17794,21267,25368,30207,35902,42600,50462,59678
%N A001522 Number of n-stacks with strictly receding walls, or planar partitions 
               of n.
%C A001522 Also number of partitions of n with positive crank (n>1), cf. A064391. 
               - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 30 2001
%D A001522 G. E. Andrews, The reasonable and unreasonable effectiveness of number 
               theory in statistical mechanics, pp. 21-34 of S. A. Burr, ed., The 
               Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. 
               Math., 46 (1992). Amer. Math. Soc.
%D A001522 G. E. Andrews, Three-quadrant Ferrers graphs, Indian J. Math., 42 (No. 
               1, 2000), 1-7.
%D A001522 F. C. Auluck, On some new types of partitions associated with generalized 
               Ferrers graphs. Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
%D A001522 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001522 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001522 E. M. Wright, Stacks, III, Quart. J. Math. Oxford, 23 (1972), 153-158.
%H A001522 T. D. Noe, <a href="b001522.txt">Table of n, a(n) for n = 0..1000</a>
%H A001522 Erich Friedman, <a href="a001522.gif">Illustration of initial terms</
               a>
%H A001522 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 A001522 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.
%F A001522 G.f.: (Sum_{k>0} -(-1)^k x^(k(k+1)/2))/(Product_{k>0} (1-x^k)).
%e A001522 For a(6)=5 we have the following stacks:
%e A001522 .x... ..x.. ...x. .xx.
%e A001522 xxxxx xxxxx xxxxx xxxx xxxxxx
%p A001522 A001522:=(1-z-z**2+z**3-z**6-2*z**7+2*z**5+z**10+z**8)/(1+z)/(z**4+z**3-1)/
               (z-1)**3; [Conjectured by S. Plouffe in his 1992 dissertation.]
%o A001522 (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)),n))
%Y A001522 a(n) = (A000041(n)-A064410(n))/2.
%Y A001522 Cf. A000041, A059776, A001523, A001524.
%Y A001522 Sequence in context: A096778 A102108 A105780 this_sequence A054405 A155167 
               A116634
%Y A001522 Adjacent sequences: A001519 A001520 A001521 this_sequence A001523 A001524 
               A001525
%K A001522 nonn,easy,nice
%O A001522 0,5
%A A001522 N. J. A. Sloane (njas(AT)research.att.com).

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