A385371 Expansion of e.g.f. 1/(1 - 2 * arcsinh(x))^(1/2).

1, 1, 3, 14, 93, 804, 8487, 105720, 1520313, 24790800, 451823403, 9101380320, 200808312405, 4816068148800, 124749498365775, 3470782979053440, 103225781141381745, 3268196553960218880, 109745731806193831635, 3895876984699452280320

Offset: 0

Author

Seiichi Manyama, Jun 27 2025

Keywords

• nonn

Crossrefs

Cf. A296675, A385372.

Cf. A001147, A385310, A385343, A385367.

Programs

PARI

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

Formula

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

a(n) = Sum_{k=0..n} A001147(k) * i^(n-k) * A385343(n,k), where i is the imaginary unit.

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