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%I #5 Feb 22 2014 06:16:20
%S 1,1,3,16,123,1221,14724,207908,3355803,60873595,1225319163,
%T 27097430328,653052022740,17036213760892,478306368143880,
%U 14381009543824236,461038595072589531,15699544671941958663,565927686301436324649
%N G.f. satisfies: A(x) = 1 + x*d/dx log(1 + x*A(x)^2).
%H Vaclav Kotesovec, <a href="/A159607/b159607.txt">Table of n, a(n) for n = 0..400</a>
%F G.f. satisfies: A(x) = 1 + x*A(x)^2*(2 - A(x)) + 2*x^2*A'(x)*A(x).
%F a(n) ~ c * n! * 2^n, where c = 0.343014753433948245763329120820010283... - _Vaclav Kotesovec_, Feb 22 2014
%e G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 123*x^4 + 1221*x^5 +...
%e A(x)^2 = 1 + 2*x + 7*x^2 + 38*x^3 + 287*x^4 + 2784*x^5 +...
%e log(1+x*A(x)^2) = x + 3*x^2/2 + 16*x^3/3 + 123*x^4/4 + 1221*x^5/5 +...
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*deriv(log(1+x*Ser(A)^2)+x*O(x^n)));polcoeff(A,n)}
%Y Cf. variants: A159606, A159608.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 16 2009
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