A280662 G.f.: Product_{k>=1, j>=1} 1/(1 - x^(j*k^4)).

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 232, 298, 387, 493, 632, 799, 1013, 1270, 1597, 1988, 2478, 3066, 3795, 4666, 5739, 7018, 8582, 10442, 12699, 15379, 18614, 22443, 27039, 32470, 38957, 46601, 55694, 66383, 79047, 93901, 111432

Offset: 0

Author

Vaclav Kotesovec, Jan 07 2017

Keywords

• nonn

Comments

In general, if m>=3 and g.f. = Product_{k>=1, j>=1} 1/(1-x^(j*k^m)), then a(n, m) ~ exp(Pi*sqrt(2*Zeta(m)*n/3) + Pi^(-1/m) * Gamma(1+1/m) * Zeta(1+1/m) * Zeta(1/m) * (6*n/Zeta(m))^(1/(2*m))) * 2^(m/4 - 7/8) * Pi^(m/4) * Zeta(m)^(1/8) / (3^(1/8) * n^(5/8)).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A006171 (m=1), A004101 (m=2), A280661 (m=3).

Cf. A280664.

Programs

Mathematica

nmax = 100; CoefficientList[Series[1/Product[1-x^(j*k^4), {k, 1, Floor[nmax^(1/4)]+1}, {j, 1, Floor[nmax/k^4]+1}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi^3 * sqrt(n/15)/3 + 2^(-7/4) * 3^(3/8) * 5^(1/8) * Pi^(-3/4) * Gamma(1/4) * Zeta(5/4) * Zeta(1/4) * n^(1/8)) * Pi^(3/2) / (3^(3/8) * 5^(1/8) * n^(5/8)).