A027999 - OEIS

%I #17 May 28 2015 03:39:07

%S 1,0,1,3,6,13,24,49,91,181,334,632,1163,2138,3880,7006,12531,22279,

%T 39369,69078,120597,209282,361405,620829,1061687,1807014,3062642,

%U 5168784,8688820,14549659,24274226,40353748,66854518,110391391,181695436,298129605,487706902

%N Expansion of Product(1+q^m)^(m(m-1)/2); m=1..inf.

%H Alois P. Heinz, <a href="/A027999/b027999.txt">Table of n, a(n) for n = 0..1000</a>

%H Vaclav Kotesovec, <a href="/A027999/a027999.jpg">Graph - the asymptotic ratio</a>

%F a(n) ~ 7^(1/8) / (2^(47/24) * 15^(1/8) * n^(5/8)) * exp(-2025 * Zeta(3)^3 / (98*Pi^8) - 135*(15/7)^(1/4) * Zeta(3)^2 / (28*Pi^5) * n^(1/4) - 3*sqrt(15/7) * Zeta(3) / (2*Pi^2) * sqrt(n) + 2*(7/15)^(1/4) * Pi/3 * n^(3/4)), where Zeta(3) = A002117. - _Vaclav Kotesovec_, May 27 2015

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p       add(binomial(binomial(i, 2), j)*b(n-i*j, i-1), j=0..n/i)))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 03 2013

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[Binomial[i, 2], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Oct 13 2014, after _Alois P. Heinz_ *)

%Y Cf. A027998, A028377, A258349, A258341, A258344.

%K nonn

%O 0,4

%A _N. J. A. Sloane_.