A304873 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).

1, 0, 14, 64, 528, 1696, 11616, 33600, 169072, 525760, 2069922, 5928066, 22259874, 59321760, 193797792, 526647420, 1566376990, 4012181104, 11456306798, 28263784110, 75995086336, 184440427360, 468750673616, 1104027571108, 2730165482640, 6239956155696

Offset

0,3

Comments

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))).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

Mathematica

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]

Crossrefs

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

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

Sequence in context: A124892 A126401 A275268 * A226754 A058092 A213757

Adjacent sequences: A304870 A304871 A304872 * A304874 A304875 A304876

Keyword

nonn

Author

Vaclav Kotesovec, May 20 2018

Status

approved