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A005986
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Number of column-strict plane partitions of n.
(Formerly M1393)
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6
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1, 2, 5, 11, 23, 45, 87, 160, 290, 512, 889, 1514, 2547, 4218, 6909, 11184, 17926, 28449, 44772, 69862, 108205, 166371, 254107, 385617, 581729, 872535, 1301722, 1932006, 2853530, 4194867, 6139361, 8946742, 12984724, 18771092, 27033892
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OFFSET
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0,2
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COMMENTS
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Note that the asymptotic formula by Gordon and Houten, cited in Stanley's paper (proposition 20.3, p. 274) is for sequence A003293, not for A005986. In addition in the same paper proposition 20.2 is wrong and Wright's formula is incomplete (for correct version see A000219). - Vaclav Kotesovec, Feb 28 2015
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: 1/Product((1-x^i)*Product(1-x^j,j=2*i-1..infinity),i=1..infinity) or 1/Product((1-x^i)^floor((i+3)/2),i=1..infinity). - Vladeta Jovovic, May 21 2006
a(n) ~ Zeta(3)^(25/72) * exp(1/24 - 25*Pi^4 / (3456*Zeta(3)) + 5*Pi^2*n^(1/3) / (24*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3)*n^(2/3) / 2) / (A^(1/2) * 2^(5/4) * 3^(1/2) * Pi * n^(61/72)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 07 2015
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MAPLE
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with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j))*b(n-j), j=1..n)/n fi end end: a:=etr(n-> `if`(modp(n, 2)=0, n+2, n+3)/2): seq(a(n), n=0..45); # Vaclav Kotesovec, Mar 02 2015 after Alois P. Heinz
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MATHEMATICA
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CoefficientList[ Series[ Product[1/((1 - x^i)*Product[(1 - x^j), {j, 2 i - 1, 40}]), {i, 40}], {x, 0, 40}], x] (* or *)
CoefficientList[ Series[ Product[1/(1 - x^j)^Floor[(j + 3)/2], {j, 40}], {x, 0, 40}], x] (* Robert G. Wilson v, May 12 2014 *)
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^((2*k+5-(-1)^k)/4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 28 2015 *)
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PROG
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(PARI) A005986_list(N, x=(O('x^N)+1)*'x)=Vec(prod(k=1, N, 1/(1-x^k)^((k+3)\2))) \\ M. F. Hasler, Sep 26 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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