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

1, 1, 3, 17, 137, 1457, 19355, 308961, 5766353, 123285153, 2972114803, 79782059249, 2360417058521, 76319622510289, 2677629295171979, 101318751122847297, 4113158120834726049, 178328823993199602241, 8223999403291995520995, 401989145900847087408849

Offset: 0

Author

Paul D. Hanna, Nov 20 2012

Keywords

• nonn

Comments

Compare to the LambertW identity: Sum_{n>=0} (n+1)^(n-1)*exp(-n*x)*x^n/n! = exp(x).

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 137*x^4/4! + 1457*x^5/5! +...
where
A(x) = 1 + sinh(x) + 3^1*sinh(x)^2/2! + 4^2*sinh(x)^3/3! + 5^3*sinh(x)^4/4! +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A003724, A136630, A162650.

Cf. A238085.

Programs

Mathematica

CoefficientList[Series[-LambertW[-Sinh[x]]/Sinh[x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 08 2013 *)

PARI

{a(n)=n!*polcoeff(sum(k=0,n,(k+1)^(k-1)*sinh(x + x*O(x^n))^k/k!),n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: LambertW(-sinh(x)) / (-sinh(x)).

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

a(n) = Sum_{k=0..n} (k+1)^(k-1) * A136630(n,k). - Seiichi Manyama, Feb 15 2025