A143740 E.g.f. satisfies A(x) = exp(x + x^2/2 * A(x)).

1, 1, 2, 7, 34, 216, 1696, 15898, 173468, 2161036, 30282076, 471599316, 8082816160, 151218316120, 3066890630168, 67031194526416, 1570793031033616, 39290173530686544, 1044871388684004304, 29440090627527552976

Offset: 0

Author

Paul D. Hanna, Aug 30 2008

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 34*x^4/4! + 216*x^5/5! + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..419
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Column k=1 of A362377.

Cf. A125500.

Programs

Mathematica

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

Maxima

a[n]:=(if n<2 then 1 else a[n-1]+sum((n-1)*(n-k)*binomial(n-2,k)*a[k]*a[n-2-k],k,0,n-2)/2);
makelist(a[n],n,0,100); /* Tani Akinari, Nov 01 2017 */

PARI

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x+x^2*A/2));(n+0)!*polcoeff(A,n)}

PARI

{a(n)=local(A=sum(m=0,n,(m+1)^(m-1)*(x^2/2)^m*exp((m+1)*x+x*O(x^n))/m!)); n!*polcoeff(A,n)}

PARI

{a(n)=local(Ex=exp(x+x*O(x^n)),W=Ex);for(k=0,n,W=exp(x*W)); n!*polcoeff(subst(W,x,x^2*Ex/2)*Ex,n)}

Formula

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

E.g.f.: A(x) = Sum_{n>=0} (n+1)^(n-1)*(x^2/2)^n*exp((n+1)*x)/n!.

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

Recurrence: a(0)=1, a(1)=1, for n > 1, a(n) = a(n-1) + Sum_{k=0..n-2} (n-1)*(n-k)*binomial(n-2,k)*a(k)*a(n-2-k)/2. - Tani Akinari, Nov 01 2017

From Seiichi Manyama, Apr 20 2023: (Start)

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

a(n) = n! * Sum_{k=0..floor(n/2)} (1/2)^k * (k+1)^(n-k-1) / (k! * (n-2*k)!). (End)