A304873 - OEIS

%I #10 May 21 2018 03:20:54

%S 1,0,14,64,528,1696,11616,33600,169072,525760,2069922,5928066,

%T 22259874,59321760,193797792,526647420,1566376990,4012181104,

%U 11456306798,28263784110,75995086336,184440427360,468750673616,1104027571108,2730165482640,6239956155696

%N G.f.: Sum_{k>=0} p(k)^4 * x^k / Sum_{k>=0} p(k)*x^k, where p(n) is the partition function A000041(n).

%C In general, if m > 1 and g.f. = Sum_{k>=0} p(k)^m * x^k / Sum_{k>=0} p(k)*x^k, then a(n, m) ~ exp(Pi*sqrt(2*(m^2 - 1)*n/3)) * ((m^2 - 1)^(m - 3/4) / (2^(2*m - 3/4) * 3^(m/2 - 1/4) * m^(2*m - 1) * n^(m - 1/4))).

%H Seiichi Manyama, <a href="/A304873/b304873.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ 2^(3/4) * 3^(3/2) * 5^(13/4) * exp(Pi*sqrt(10*n)) / (2^22 * n^(15/4)).

%t nmax = 25; CoefficientList[Series[Sum[PartitionsP[k]^4*x^k, {k, 0, nmax}] / Sum[PartitionsP[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

%Y Cf. A054440 (m=2), A260664 (m=3).

%Y Cf. A000041, A001255, A133042, A304877, A304878.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, May 20 2018