A383315 -id:A383315 - cp's OEIS frontend

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

Seiichi Manyama, Apr 23 2025

nonn

Cf. A383314, A383315.

Cf. A002801.

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

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

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

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 23 2025

nonn

Cf. A383313, A383315.

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

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

cp's OEIS Frontend

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

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