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
Keywords
Examples
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! +...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..100
Crossrefs
Cf. A277464.
Programs
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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 *) -
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), ", "))
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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
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
Extensions
E.g.f. in Formula section corrected by Paul D. Hanna, Jul 08 2013, error noted by Vaclav Kotesovec.
Comments