A304992 G.f.: Sum_{k>=0} A000041(k)^3 * x^k / Sum_{k>=0} A000009(k) * x^k.

1, 0, 7, 18, 98, 210, 969, 1938, 7037, 15258, 44815, 93180, 262391, 518550, 1311015, 2657328, 6189160, 12124098, 27239760, 52063668, 111630480, 211503288, 432900236, 806091180, 1610854427, 2940167268, 5691072911, 10289144976, 19402974147, 34523231688

Offset: 0

Author

Vaclav Kotesovec, May 23 2018

Keywords

• nonn

Comments

In general, if m > 1 and g.f. = Sum_{k>=0} A000041(k)^m * x^k / Sum_{k>=0} A000009(k) * x^k, then a(n, m) ~ exp(Pi*sqrt((2*m^2 - 1)*n/3)) * ((2*m^2 - 1)^(m - 1/2) / (2^(3*m - 1) * 3^(m/2) * m^(2*m - 1) * n^m)).

Crossrefs

Cf. A000009, A000041, A133042, A260664, A304873, A304878, A304988.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Sum[PartitionsP[k]^3*x^k, {k, 0, nmax}] / Sum[PartitionsQ[k]*x^k, {k, 0, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ 289 * sqrt(17/3) * exp(Pi*sqrt(17*n/3)) / (186624*n^3).