%I #14 Jan 02 2013 21:08:04

%S 1,1,4,30,372,6865,180337,6664746,347126163,25621940636,2692407417070,

%T 403782925417406,86521373592454354,26501533097131820625,

%U 11604995872798575186212,7265288737454158428096319,6503007077048714757847413184,8322734168520894445448609153627

%N G.f. satisfies: A(x) = x*Sum{n>=0} A( n*x/(1+n*x) )^n.

%C Compare g.f. to the identity (cf. A001710):

%C Sum_{n>=0} ( n*x/(1+n*x) )^n = 1 + Sum_{n>=1} (n+1)!/2 * x^n.

%C Also compare to the identity:

%C G(x) = x*Sum_{n>=0} G( x/(1+x) )^n is satisfied by: G(x) = x/(1-x).

%e G.f.: A(x) = x + x^2 + 4*x^3 + 30*x^4 + 372*x^5 + 6865*x^6 +...

%e where

%e A(x) = x + x*A(x/(1+x)) + x*A(2*x/(1+2*x))^2 + x*A(3*x/(1+3*x))^3 + x*A(4*x/(1+4*x))^4 + x*A(5*x/(1+5*x))^5 +...

%o (PARI) {a(n)=local(A=x);for(i=1,n,A=x*sum(m=0,n,subst(A,x,m*x/(1+m*x+x*O(x^n)))^m));polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jan 02 2013