A385308 Expansion of e.g.f. 1/(1 - 2 * x * cosh(x))^(1/2).

1, 1, 3, 18, 141, 1400, 17055, 245392, 4070073, 76483584, 1606033755, 37267953536, 947051118981, 26156846230528, 780174007426359, 24992424003517440, 855795857724702705, 31193844533488074752, 1205893835653392258867, 49280187764171870470144, 2122704756621224015194365

Offset: 0

Author

Seiichi Manyama, Jun 24 2025

Keywords

• nonn

Crossrefs

Cf. A205571, A385309.

Cf. A380015, A385304, A385306, A385310.

Cf. A001147, A185951, A352648, A385281.

Programs

PARI

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*a185951(n, k));

Formula

a(n) = Sum_{k=0..n} A001147(k) * A185951(n,k), where A185951(n,0) = 0^n.

a(n) ~ sqrt(2) * n^n / (sqrt(1 + r*sqrt(1 - 4*r^2)) * exp(n) * r^n), where r = 0.452787214835453627588998503316635625709288535855... is the root of the equation 2*r*cosh(r) = 1. - Vaclav Kotesovec, Jun 28 2025