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G.f. satisfies: A(x) = d/dx log(1+x + x^2*A(x)^2).
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%I #14 Aug 24 2017 06:58:51

%S 1,1,4,29,286,3478,49750,813949,14951218,304272526,6791813216,

%T 164961150626,4331176071496,122252442659992,3692061705866554,

%U 118804126659832861,4058311931802683890,146684121878245070758,5593222574333657589416,224400536392956665317414

%N G.f. satisfies: A(x) = d/dx log(1+x + x^2*A(x)^2).

%H Vaclav Kotesovec, <a href="/A182356/b182356.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. satisfies: A(x) = (1 + 2*x*A(x)^2 - x^2*A(x)^3) / (1+x - 2*x^2*A'(x)).

%F a(n) ~ c * n * 2^n * n!, where c = 0.1840416364326449945692... - _Vaclav Kotesovec_, Aug 24 2017

%e G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 286*x^4 + 3478*x^5 + 49750*x^6 +...

%e such that

%e log(1+x + x^2*A(x)^2) = x + x^2/2 + 4*x^3/3 + 29*x^4/4 + 286*x^5/5 + 3478*x^6/6 + 49750*x^7/7 +...+ a(n-1)*x^n/n +...

%e Related expansions.

%e 1+x + x^2*A(x)^2 = 1 + x + x^2 + 2*x^3 + 9*x^4 + 66*x^5 + 646*x^6 + 7760*x^7 109585*x^8 +...+ A259607(n)*x^n +...

%e A(x)^2 = 1 + 2*x + 9*x^2 + 66*x^3 + 646*x^4 + 7760*x^5 + 109585*x^6 +...+ A259607(n+2)*x^n +...

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=deriv(log(1+x + x^2*A^2 +x^2*O(x^n))));polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A259607.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 26 2012