A381280 Expansion of e.g.f. 1/(1 - x * cosh(2*x)).

1, 1, 2, 18, 120, 920, 10320, 126448, 1714048, 27073152, 472354560, 8989147904, 187690331136, 4245706716160, 103239264593920, 2691918892861440, 74885151106498560, 2212607133043884032, 69227613551324233728, 2286465386258267176960, 79487593489348266557440

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..20.

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).

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

a(n) ~ sqrt(Pi) * 2^(n + 5/2) * n^(n + 1/2) / ((1 + sinh(r))^2 * exp(n) * r^(n+2)), where r = A201939. - Vaclav Kotesovec, Apr 19 2025

Mathematica

With[{nn=20}, CoefficientList[Series[1/(1-x Cosh[2x]), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jan 27 2026 *)

Prog

(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)*a185951(n, k));

Crossrefs

Cf. A205571, A381281.

Cf. A185951, A201939.

Sequence in context: A058052 A119578 A052610 * A052653 A342124 A289830

Adjacent sequences: A381277 A381278 A381279 * A381281 A381282 A381283

Keyword

nonn

Author

Seiichi Manyama, Feb 18 2025

Status

approved