A218296 Expansion of e.g.f. Sum_{n>=0} n^n * cosh(n*x) * x^n/n!.

1, 1, 4, 30, 352, 5560, 109056, 2540720, 68401152, 2087897472, 71236526080, 2686375597312, 110951893303296, 4980913763830784, 241491517062512640, 12575483733378816000, 700015678015053758464, 41480146826887546372096, 2606901492484549499682816

Offset

0,3

Comments

Compare the e.g.f. to the identity: Sum_{n>=0} n^n * exp(-n*x) * x^n/n! = 1/(1-x).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Formula

E.g.f.: 1 + (1/2)*x/(1-x) - (1/2)*LambertW(-x*exp(x))/(1 + LambertW(-x*exp(x))).

a(n) ~ n^n/(2*sqrt(1+LambertW(exp(-1)))*exp(n)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Jul 08 2013

a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^n * binomial(n,2*k). - Seiichi Manyama, Feb 15 2023

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 30*x^3/3! + 352*x^4/4! + 5560*x^5/5! +...
where
A(x) = 1 + 1^1*x*cosh(1*x) + 2^2*cosh(2*x)*x^2/2! + 3^3*cosh(3*x)*x^3/3! + 4^4*cosh(4*x)*x^4/4! + 5^5*cosh(5*x)*x^5/5! +...

Mathematica

CoefficientList[Series[1 + 1/2*x/(1-x) - 1/2*LambertW[-x*E^x]/(1 + LambertW[-x*E^x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 08 2013 *)

Prog

(PARI) a(n)=n!*polcoeff(sum(k=0, n, k^k*cosh(k*x +x*O(x^n))*x^k/k!), n)
for(n=0, 30, print1(a(n), ", "))
(PARI) LambertW(x, N)=sum(n=1, N, (-n)^(n-1)*x^n/n!)
{a(n)=local(X=x+x*O(x^n)); n!*polcoeff(1 + (1/2)*x/(1-X) - (1/2)*LambertW(-x*exp(X), n)/(1 + LambertW(-x*exp(X), n)), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 08 2013
(PARI) a(n) = sum(k=0, n\2, (n-2*k)^n*binomial(n, 2*k)); \\ Seiichi Manyama, Feb 15 2023

Crossrefs

Cf. A277464.

Sequence in context: A052316 A089918 A132622 * A118792 A317030 A192549

Adjacent sequences: A218293 A218294 A218295 * A218297 A218298 A218299

Keyword

nonn

Author

Paul D. Hanna, Oct 27 2012

Extensions

E.g.f. in Formula section corrected by Paul D. Hanna, Jul 08 2013, error noted by Vaclav Kotesovec.

Status

approved