A280664 G.f.: Product_{k>=1, j>=1} (1 + x^(j*k^4)).

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 66, 79, 93, 109, 128, 150, 175, 204, 237, 274, 318, 367, 423, 487, 559, 641, 734, 839, 957, 1091, 1241, 1410, 1601, 1814, 2053, 2322, 2622, 2957, 3334, 3752, 4218, 4740, 5318, 5962, 6679

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+x^(j*k^m)), then a(n, m) ~ exp(Pi*sqrt(Zeta(m)*n/3) + (2^(1/m)-1) * Pi^(-1/m) * Gamma(1+1/m) * Zeta(1+1/m) * Zeta(1/m) * (3*n/Zeta(m))^(1/(2*m))) * Zeta(m)^(1/4) / (2^(5/4) * 3^(1/4) * n^(3/4)).

Links

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

Crossrefs

Cf. A107742 (m=1), A280451 (m=2), A280663 (m=3).

Cf. A280662.

Programs

Mathematica

nmax = 100; CoefficientList[Series[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/30)/3 + 2^(-15/8) * 3^(3/8) * 5^(1/8) * (2^(1/4)-1) * Pi^(-3/4) * Gamma(1/4) * Zeta(5/4) * Zeta(1/4) * n^(1/8)) * Pi / (2^(3/2) * 3^(3/4) * 5^(1/4) * n^(3/4)).