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A218296
Expansion of e.g.f. Sum_{n>=0} n^n * cosh(n*x) * x^n/n!.
2
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
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: A089918 A371041 A132622 * A118792 A317030 A192549
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