A381344 Expansion of e.g.f. 1/( 1 - x * cosh(sqrt(2)*x) ).

1, 1, 2, 12, 72, 500, 4560, 47936, 565376, 7572240, 112838400, 1844425792, 32910332928, 636463467328, 13251265570816, 295598326909440, 7034150340034560, 177843592245969152, 4760839037033054208, 134528586280018721792, 4001489050575059025920, 124973219149863342633984

Offset: 0

Author

Seiichi Manyama, Feb 20 2025

Keywords

• nonn

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A352252, A381280, A381281, A381282, A381283, A381345.

Cf. A185951.

Programs

PARI

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*2^((n-k)/2)*a185951(n, k));

Formula

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

a(n) = Sum_{k=0..n} k! * 2^((n-k)/2) * A185951(n,k).

a(n) ~ sqrt(Pi) * 2^(n/2 + 1) * n^(n + 1/2) / ((cosh(r) + r*sinh(r)) * exp(n) * r^(n+1)), where r = 0.95090803593755778120914299086438615849657408871... is the root of the equation r*cosh(r) = sqrt(2). - Vaclav Kotesovec, Apr 19 2025