A162650 E.g.f. satisfies: A(x) = exp( sinh(x*A(x)) ).

1, 1, 3, 17, 145, 1657, 23779, 411329, 8335233, 193742001, 5082443619, 148557955537, 4788293136081, 168731904680809, 6453754733373315, 266292315945201409, 11790632930559453313, 557631368540974947169

Offset: 0

Author

Paul D. Hanna, Jul 08 2009

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 145*x^4/4! + 1657*x^5/5! +...
log(A(x)) = sinh(x*A(x)) where
log(A(x)) = x + 2*x^2/2! + 10*x^3/3! + 80*x^4/4! + 876*x^5/5! + 12192*x^6/6! +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..370

Crossrefs

Cf. A162649.

Programs

PARI

{a(n,m=1)=sum(k=0,n,m*(n+m)^(k-1)/k!*sum(j=0,k,(-1)^(k-j)*binomial(k,j)*(2*j-k)^n/2^k))}

Formula

a(n) = Sum_{k=0..n} (n+1)^(k-1)/k! * Sum_{j=0..k} C(k,j)*(2j-k)^n/2^k.

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then

a(n,m) = Sum_{k=0..n} m*(n+m)^(k-1)/k! * Sum_{j=0..k} C(k,j)*(2j-k)^n/2^k.

a(n) ~ n^(n-1) / (sqrt(1/s^2+r^2*log(s)) * exp(n) * r^n), where r = 0.3296546568511367672... and s = 2.320640520774412031... are roots of the system of equations sinh(r*s) = log(s), r*s*cosh(r*s) = 1. - Vaclav Kotesovec, Jul 15 2014