A385422 Expansion of e.g.f. 1/(1 - arcsin(3*x))^(1/3).

1, 1, 4, 37, 424, 6889, 129376, 3004597, 78196864, 2363157937, 78520720384, 2924352594373, 118146438461440, 5232528466643737, 248845526415892480, 12778931460471237397, 699044652076991610880, 40846771050451091426785, 2526020027235443981025280

Offset: 0

Author

Seiichi Manyama, Jun 28 2025

Keywords

• nonn

Crossrefs

Cf. A189780, A385421.

Cf. A007559, A007788, A235135, A385343, A385420.

Programs

PARI

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

Formula

a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A385343(n,k).

a(n) ~ sqrt(2*Pi) * cos(1)^(1/3) * 3^n * n^(n - 1/6) / (Gamma(1/3) * sin(1)^(n + 1/3) * exp(n)). - Vaclav Kotesovec, Jun 28 2025