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A001522
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Number of n-stacks with strictly receding walls, or planar partitions of n.
(Formerly M0644 N0238)
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7
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0, 1, 1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 35, 47, 62, 82, 107, 139, 179, 230, 293, 372, 470, 591, 740, 924, 1148, 1422, 1756, 2161, 2651, 3244, 3957, 4815, 5844, 7075, 8545, 10299, 12383, 14859, 17794, 21267, 25368, 30207, 35902, 42600, 50462, 59678
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Also number of partitions of n with positive crank (n>1), cf. A064391. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 30 2001
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REFERENCES
| 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.
G. E. Andrews, Three-quadrant Ferrers graphs, Indian J. Math., 42 (No. 1, 2000), 1-7.
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.
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
Erich Friedman, Illustration of initial terms
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| G.f.: (Sum_{k>0} -(-1)^k x^(k(k+1)/2))/(Product_{k>0} (1-x^k)).
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EXAMPLE
| For a(6)=5 we have the following stacks:
.x... ..x.. ...x. .xx.
xxxxx xxxxx xxxxx xxxx xxxxxx
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MAPLE
| 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.]
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MATHEMATICA
| max = 50; f[x_] := Sum[-(-1)^k*x^(k*(k+1)/2), {k, 1, max}] / Product[(1-x^k), {k, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* From Jean-François Alcover, Dec 27 2011, 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)), n))
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CROSSREFS
| a(n) = (A000041(n)-A064410(n))/2.
Cf. A000041, A059776, A001523, A001524.
Sequence in context: A096778 A102108 A105780 * A054405 A155167 A116634
Adjacent sequences: A001519 A001520 A001521 * A001523 A001524 A001525
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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