A383314 Expansion of e.g.f. exp(-x/2) / (1-4*x)^(1/8).

1, 0, 2, 16, 204, 3392, 69880, 1717824, 49077392, 1597961728, 58410015264, 2368359845120, 105492853521088, 5120497605295104, 269008689666893696, 15207860554294309888, 920541893947665404160, 59401332750388003782656, 4070589051420604880962048

Offset: 0

Author

Seiichi Manyama, Apr 23 2025

Keywords

• nonn

Crossrefs

Cf. A383313, A383315.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x/2)/(1-4*x)^(1/8)))

Formula

a(n) = (-1)^n * n! * Sum_{k=0..n} (1/2)^(n-3*k) * binomial(-1/8,k)/(n-k)!.

a(n) = 2*(n-1) * (2*a(n-1) + a(n-2)) for n > 1.

a(n) ~ sqrt(Pi) * 2^(2*n + 1/2) * n^(n - 3/8) / (Gamma(1/8) * exp(n + 1/8)). - Vaclav Kotesovec, Apr 23 2025