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%I #17 May 03 2015 07:07:34
%S 1,1,4,26,219,2227,26438,359904,5555201,96383191,1864908541,
%T 39929905561,938897407239,24069888638463,668309231078015,
%U 19977542570492051,639571311256259372,21828488143257352752,791044181963746918758,30331001954496565907536
%N G.f. satisfies: A(x) = Sum_{n>=0} ((1 + x*A(x))^n - 1)^n / (1 + x*A(x))^(n^2).
%C Limit n->infinity A220353(n)/A187826(n) = 1. - _Vaclav Kotesovec_, Nov 08 2014
%H Vaclav Kotesovec, <a href="/A187826/b187826.txt">Table of n, a(n) for n = 0..140</a>
%F G.f. satisfies: A(x) = Sum_{n>=1} ((1+x*A(x))^n - 1)^(n-1) / (1+x*A(x))^(n^2).
%F a(n) ~ c * n^n / (exp(n) * (log(2))^(2*n)), where c = 2.341658334181687683758... . - _Vaclav Kotesovec_, Nov 08 2014
%F In closed form, c = 1 / (log(2) * sqrt(1-log(2)) * 2^((1-log(2))/2)). - _Vaclav Kotesovec_, May 03 2015
%e G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 219*x^4 + 2227*x^5 + 26438*x^6 +...
%e where the g.f. satisfies the identities:
%e (1) A(x) = 1 + x*A(x)/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)^2/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^3/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^4/(1+x*A(x))^16 +...
%e (2) A(x) = 1/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^2/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^3/(1+x*A(x))^16 +...
%o (PARI) {a(n)=local(q, A=1); for(i=1,n,q=1+x*A+x*O(x^n);A=sum(k=0, n+1, q^(-k^2)*(q^k-1)^k)); polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n)=local(q, A=1); for(i=1,n,q=1+x*A+x*O(x^n);A=sum(k=1, n+1, q^(-k^2)*(q^k-1)^(k-1))); polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A220352, A187827, A220353.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 27 2012
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