A352648 Expansion of e.g.f. 1/(1 - 2 * x * cosh(x)).

1, 2, 8, 54, 480, 5290, 70080, 1083614, 19145728, 380552274, 8404669440, 204182993542, 5411361939456, 155365918497530, 4803852288901120, 159142710151610670, 5623576097060290560, 211138456468635968674, 8393550198348236193792, 352212802264773650385110

Offset: 0

Author

Seiichi Manyama, Mar 25 2022

Keywords

• nonn

Crossrefs

Cf. A205571, A352649.

Cf. A185951.

Programs

Mathematica

With[{m = 19}, Range[0, m]! * CoefficientList[Series[1/(1 - 2*x*Cosh[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

PARI

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

PARI

a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (2*k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).

a(n) ~ n! / ((1 + r * sqrt(1 - 4*r^2)) * r^n), where r = 0.452787214835453627588998503316635625709288535855800416726... is the root of the equation 2*r*cosh(r) = 1. - Vaclav Kotesovec, Mar 27 2022

a(n) = Sum_{k=0..n} 2^k * k! * A185951(n,k). - Seiichi Manyama, Jun 25 2025