%I #6 Feb 07 2013 02:51:26
%S 1,1,2,7,38,302,3428,55083,1251590,40289986,1841412556,119672298150,
%T 11071253179356,1459246211179612,274215745471606536,
%U 73511068056751643571,28128768433558172885958,15371204139970896651788090,12001328910786418412379456956
%N G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * Product_{k=1..n} A(k*x)/(1 + k*x*A(k*x)).
%C Compare to the identity: 1/(1-x) = Sum_{n>=0} n!*x^n/Product_{k=1..n} (1+k*x).
%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 38*x^4 + 302*x^5 + 3428*x^6 +...
%e where, by definition,
%e A(x) = 1 + x*A(x)/(1+x*A(x)) + 2!*x^2*A(x)*A(2*x)/((1+x*A(x))*(1+2*x*A(2*x))) + 3!*x^3*A(x)*A(2*x)*A(3*x)/((1+x*A(x))*(1+2*x*A(2*x))*(1+3*x*A(3*x))) + 4!*x^4*A(x)*A(2*x)*A(3*x)*A(4*x)/((1+x*A(x))*(1+2*x*A(2*x))*(1+3*x*A(3*x))*(1+4*x*A(4*x))) +...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m!*x^m*prod(k=1, m, subst(A,x,k*x+x*O(x^n ))/(1+k*x*subst(A,x,k*x+x*O(x^n)))))); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 06 2013
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