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

%I A218296 #31 Feb 19 2023 13:06:07
%S A218296 1,1,4,30,352,5560,109056,2540720,68401152,2087897472,71236526080,
%T A218296 2686375597312,110951893303296,4980913763830784,241491517062512640,
%U A218296 12575483733378816000,700015678015053758464,41480146826887546372096,2606901492484549499682816
%N A218296 Expansion of e.g.f. Sum_{n>=0} n^n * cosh(n*x) * x^n/n!.
%C A218296 Compare the e.g.f. to the identity: Sum_{n>=0} n^n * exp(-n*x) * x^n/n! = 1/(1-x).
%H A218296 Vincenzo Librandi, <a href="/A218296/b218296.txt">Table of n, a(n) for n = 0..100</a>
%F A218296 E.g.f.: 1 + (1/2)*x/(1-x) - (1/2)*LambertW(-x*exp(x))/(1 + LambertW(-x*exp(x))).
%F A218296 a(n) ~ n^n/(2*sqrt(1+LambertW(exp(-1)))*exp(n)*LambertW(exp(-1))^n). - _Vaclav Kotesovec_, Jul 08 2013
%F A218296 a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^n * binomial(n,2*k). -  _Seiichi Manyama_, Feb 15 2023
%e A218296 E.g.f.: A(x) = 1 + x + 4*x^2/2! + 30*x^3/3! + 352*x^4/4! + 5560*x^5/5! +...
%e A218296 where
%e A218296 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! +...
%t A218296 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 *)
%o A218296 (PARI) a(n)=n!*polcoeff(sum(k=0, n, k^k*cosh(k*x +x*O(x^n))*x^k/k!), n)
%o A218296 for(n=0, 30, print1(a(n), ", "))
%o A218296 (PARI) LambertW(x,N)=sum(n=1,N,(-n)^(n-1)*x^n/n!)
%o A218296 {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)}
%o A218296 for(n=0, 30, print1(a(n), ", ")) \\ _Paul D. Hanna_, Jul 08 2013
%o A218296 (PARI) a(n) = sum(k=0, n\2, (n-2*k)^n*binomial(n, 2*k)); \\ _Seiichi Manyama_, Feb 15 2023
%Y A218296 Cf. A277464.
%K A218296 nonn
%O A218296 0,3
%A A218296 _Paul D. Hanna_, Oct 27 2012
%E A218296 E.g.f. in Formula section corrected by _Paul D. Hanna_, Jul 08 2013, error noted by _Vaclav Kotesovec_.