A161631 E.g.f. satisfies A(x) = 1 + x*exp(x*A(x)).

1, 1, 2, 9, 52, 425, 4206, 50827, 713000, 11500785, 208833850, 4226139731, 94226705772, 2296472176297, 60727113115046, 1732020500240955, 52998549321251536, 1731977581804704737, 60205422811336194546

Offset: 0

Author

Paul D. Hanna, Jun 18 2009

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 425*x^5/5! +...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..402

Crossrefs

Cf. A125500, A364978, A364979.

Programs

Mathematica

CoefficientList[Series[1-LambertW[-x^2*E^x]/x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)

PARI

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

Formula

E.g.f.: A(x) = 1 - LambertW(-x^2*exp(x))/x.

E.g.f.: A(x) = 1 + Sum_{n>=1} x^(2*n-1) * n^(n-1) * exp(n*x) / n!.

E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = 1 + x*exp(x)/B(x) = (1+sqrt(1+4*x*exp(x)))/2.

a(n) = n*A125500(n-1) for n>0, where exp(x*A(x)) = e.g.f. of A125500.

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

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

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

a(n) ~ sqrt(1+LambertW(1/(2*exp(1/2)))) * n^(n-1) / (exp(n) * 2^(n+1/2) * (LambertW(1/(2*exp(1/2))))^(n+1)). - Vaclav Kotesovec, Jul 09 2013