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%I M1393 #47 Sep 27 2018 12:30:57
%S 1,2,5,11,23,45,87,160,290,512,889,1514,2547,4218,6909,11184,17926,
%T 28449,44772,69862,108205,166371,254107,385617,581729,872535,1301722,
%U 1932006,2853530,4194867,6139361,8946742,12984724,18771092,27033892
%N Number of column-strict plane partitions of n.
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005986/b005986.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="/A005986/a005986.jpg">Graph - The asymptotic ratio</a>.
%H Richard P. Stanley, <a href="http://www-math.mit.edu/~rstan//pubs/pubfiles/12-2.pdf">Theory and Application of Plane Partitions, II</a>, Studies in Appl. Math. 50 (1971), 259-279. <a href="http://doi.org/10.1002/sapm1971503259">DOI:10.1002/sapm1971503259</a>.
%F 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
%F 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
%p 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_
%t CoefficientList[ Series[ Product[1/((1 - x^i)*Product[(1 - x^j), {j, 2 i - 1, 40}]), {i, 40}], {x, 0, 40}], x] (* or *)
%t CoefficientList[ Series[ Product[1/(1 - x^j)^Floor[(j + 3)/2], {j, 40}], {x, 0, 40}], x] (* _Robert G. Wilson v_, May 12 2014 *)
%t 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 *)
%o (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
%Y Cf. A003293.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, May 21 2006
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