A385282 Expansion of e.g.f. 1/(1 - 3 * x * cosh(3*x))^(1/3).

1, 1, 4, 55, 712, 11605, 248320, 6218443, 178519936, 5846857993, 214490045440, 8700546508159, 387053184719872, 18737207168958109, 980424546959183872, 55142056940797803475, 3317502712746788945920, 212592531182720568805777, 14456626429227650204041216

Offset: 0

Author

Seiichi Manyama, Jun 24 2025

Keywords

• nonn

Crossrefs

Cf. A205571, A385281.

Cf. A007559, A185951.

Cf. A069814.

Programs

PARI

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*3^(n-k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A007559(k) * 3^(n-k) * A185951(n,k), where A185951(n,0) = 0^n.

a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 1/6) / (Gamma(1/3) * (1/r + sqrt(1 - r^2))^(1/3) * exp(n) * r^(n + 1/3)), where r = A069814. - Vaclav Kotesovec, Jun 24 2025