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A219503 Expansion of e.g.f. Sum_{n>=0} (n+1)^(n-1) * sinh(x)^n / n!.

%I A219503 #19 Feb 15 2025 10:11:40
%S A219503 1,1,3,17,137,1457,19355,308961,5766353,123285153,2972114803,
%T A219503 79782059249,2360417058521,76319622510289,2677629295171979,
%U A219503 101318751122847297,4113158120834726049,178328823993199602241,8223999403291995520995,401989145900847087408849
%N A219503 Expansion of e.g.f. Sum_{n>=0} (n+1)^(n-1) * sinh(x)^n / n!.
%C A219503 Compare to the LambertW identity: Sum_{n>=0} (n+1)^(n-1)*exp(-n*x)*x^n/n! = exp(x).
%H A219503 Vincenzo Librandi, <a href="/A219503/b219503.txt">Table of n, a(n) for n = 0..100</a>
%F A219503 E.g.f.: LambertW(-sinh(x)) / (-sinh(x)).
%F A219503 a(n) ~ (1+exp(2))^(1/4) * n^(n-1) / (exp(n-1) * log(exp(-1) +sqrt(1+exp(-2)))^(n-1/2)). - _Vaclav Kotesovec_, Jul 08 2013
%F A219503 a(n) = Sum_{k=0..n} (k+1)^(k-1) * A136630(n,k). - _Seiichi Manyama_, Feb 15 2025
%e A219503 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 137*x^4/4! + 1457*x^5/5! +...
%e A219503 where
%e A219503 A(x) = 1 + sinh(x) + 3^1*sinh(x)^2/2! + 4^2*sinh(x)^3/3! + 5^3*sinh(x)^4/4! +...
%t A219503 CoefficientList[Series[-LambertW[-Sinh[x]]/Sinh[x], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jul 08 2013 *)
%o A219503 (PARI) {a(n)=n!*polcoeff(sum(k=0,n,(k+1)^(k-1)*sinh(x + x*O(x^n))^k/k!),n)}
%o A219503 for(n=0,25,print1(a(n),", "))
%Y A219503 Cf. A003724, A136630, A162650.
%Y A219503 Cf. A238085.
%K A219503 nonn
%O A219503 0,3
%A A219503 _Paul D. Hanna_, Nov 20 2012