A385372 Expansion of e.g.f. 1/(1 - 3 * arcsinh(x))^(1/3).

1, 1, 4, 27, 264, 3369, 52896, 986187, 21293184, 522491697, 14359993344, 436964488443, 14583637923840, 529683272760537, 20798444046458880, 877927319167721067, 39644175780617748480, 1906959640776766940385, 97344936393086594580480, 5255894631271228490720475

Offset: 0

Author

Seiichi Manyama, Jun 27 2025

Keywords

• nonn

Crossrefs

Cf. A296675, A385371.

Cf. A007559, A385311, A385343.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*asinh(x))^(1/3)))

Formula

E.g.f.: 1/(1 - 3 * log(x + sqrt(x^2 + 1)))^(1/3).

a(n) = Sum_{k=0..n} A007559(k) * i^(n-k) * A385343(n,k), where i is the imaginary unit.

a(n) ~ sqrt(Pi) * (exp(2/3) + 1)^(1/3) * 2^(n + 1/2) * n^(n - 1/6) / (3^(1/3) * Gamma(1/3) * exp(2*n/3) * (exp(2/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Jun 27 2025