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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 2 04:17 EST 2023. Contains 367506 sequences. (Running on oeis4.)