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A005986 Number of column-strict plane partitions of n.

%I A005986 M1393 #47 Sep 27 2018 12:30:57
%S A005986 1,2,5,11,23,45,87,160,290,512,889,1514,2547,4218,6909,11184,17926,
%T A005986 28449,44772,69862,108205,166371,254107,385617,581729,872535,1301722,
%U A005986 1932006,2853530,4194867,6139361,8946742,12984724,18771092,27033892
%N A005986 Number of column-strict plane partitions of n.
%C A005986 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 A005986 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005986 Vincenzo Librandi, <a href="/A005986/b005986.txt">Table of n, a(n) for n = 0..1000</a>
%H A005986 Vaclav Kotesovec, <a href="/A005986/a005986.jpg">Graph - The asymptotic ratio</a>.
%H A005986 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 A005986 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 A005986 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 A005986 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 A005986 CoefficientList[ Series[ Product[1/((1 - x^i)*Product[(1 - x^j), {j, 2 i - 1, 40}]), {i, 40}], {x, 0, 40}], x] (* or *)
%t A005986 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 A005986 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 A005986 (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 A005986 Cf. A003293.
%K A005986 nonn
%O A005986 0,2
%A A005986 _N. J. A. Sloane_
%E A005986 More terms from _Vladeta Jovovic_, May 21 2006