A383313 Expansion of e.g.f. exp(-x/2) / (1-2*x)^(1/4).

1, 0, 1, 4, 27, 232, 2455, 30852, 449113, 7432624, 137829249, 2830911220, 63796168579, 1565078980536, 41521403685463, 1184510408920468, 36158133322895985, 1176012432875399008, 40599110984252798017, 1482736219224857910756, 57115359439245403771051

Offset: 0

Author

Seiichi Manyama, Apr 23 2025

Keywords

• nonn

Crossrefs

Cf. A383314, A383315.

Cf. A002801.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x/2)/(1-2*x)^(1/4)))

Formula

a(n) = (-1)^n * n! * Sum_{k=0..n} (1/2)^(n-2*k) * binomial(-1/4,k)/(n-k)!.

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

a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 1/4) / (Gamma(1/4) * exp(n + 1/4)). - Vaclav Kotesovec, Apr 23 2025