A383316 Expansion of e.g.f. exp(x/2) / (1-4*x)^(1/8).

1, 1, 3, 23, 281, 4593, 93643, 2285959, 64981809, 2107824353, 76819828499, 3107456481399, 138145505435977, 6694550810809297, 351219409831557339, 19832058937696108007, 1199219012904515868257, 77314609952787255980481, 5293934640303567123132451

Offset: 0

Author

Seiichi Manyama, Apr 23 2025

Keywords

• nonn

Crossrefs

Cf. A002801, A383317.

Programs

PARI

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

Formula

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

a(n) = (4*n-3)*a(n-1) - 2*(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