A383627 Expansion of 1/( Product_{k=0..2} (1 - (3*k+1) * x) )^(1/3).

1, 4, 19, 100, 562, 3304, 20062, 124744, 789553, 5065444, 32840347, 214681636, 1412786872, 9348241504, 62138211112, 414627600736, 2775808278058, 18636412183336, 125436195473662, 846145250012776, 5719044971926972, 38723124875350960, 262609593669266404

Offset: 0

Author

Seiichi Manyama, May 03 2025

Keywords

• nonn

Comments

In general, if m > 0 and g.f. = 1/(Product_{k=0..m-1} (1 - (m*k+1)*x))^(1/m), then a(n) ~ (m*(m-1) + 1)^(n + 1 - 1/m) / (Gamma(1/m) * Gamma(m+1)^(1/m) * m^(1 - 2/m) * n^(1 - 1/m)). - Vaclav Kotesovec, Aug 18 2025

Crossrefs

Cf. A383628, A383629, A383630, A383631, A383632, A383633.

Cf. A016223, A370781, A383935.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(1/prod(k=0, 2, 1-(3*k+1)*x)^(1/3))

Formula

a(n) ~ 7^(n + 2/3) / (Gamma(1/3) * 2^(1/3) * 3^(2/3) * n^(2/3)). - Vaclav Kotesovec, May 12 2025

a(n) = Sum_{k=0..n} binomial(n,k) * A383935(k). - Seiichi Manyama, Aug 18 2025