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A137974 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^6.

%I A137974 #13 Mar 03 2018 12:28:43
%S A137974 1,1,6,45,410,4020,41826,452207,5033910,57300285,663912420,7804131660,
%T A137974 92838682242,1115595461915,13521340799310,165104951405235,
%U A137974 2029162664033790,25081468301798301,311593507408597920
%N A137974 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^6.
%C A137974 In general, if g.f. satisfies A(x) = 1 + x*(1 + x*A(x)^p)^q, p >= 1, q >= 1, p + q > 2, then a(n) ~ sqrt(q*s*(1-s)*(p*(1-s)-s) / (2*Pi*p*(q-s-p*q*(1-s)))) / (n^(3/2) * r^n), where r and s are real roots of the system of equations s = 1 + r*(1 + r*s^p)^q, p*q * r^2 * s^(p-1) * (1 + r*s^p)^(q-1) = 1. - _Vaclav Kotesovec_, Nov 22 2017
%H A137974 Vaclav Kotesovec, <a href="/A137974/b137974.txt">Table of n, a(n) for n = 0..300</a>
%F A137974 G.f.: A(x) = 1 + x*B(x)^6 where B(x) is the g.f. of A137973.
%F A137974 a(n) = Sum_{k=0..n-1} C(6*(n-k),k)/(n-k) * C(5*k,n-k-1) for n>0 with a(0)=1. [_Paul D. Hanna_, Jun 16 2009]
%F A137974 a(n) ~ sqrt(6*s*(1-s)*(5-6*s) / ((290*s - 240)*Pi)) / (n^(3/2) * r^n), where r = 0.0739607593319208338998816978154858830062403258604... and s = 1.234938729532398384561936758596402363403570701060... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^6, 30 * r^2 * s^4 * (1 + r*s^5)^5 = 1. - _Vaclav Kotesovec_, Nov 22 2017
%o A137974 (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^6);polcoeff(A,n)}
%o A137974 (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(n-k),k)/(n-k)*binomial(5*k,n-k-1))) \\ _Paul D. Hanna_, Jun 16 2009
%Y A137974 Cf. A137973; A137968, A137970, A137972.
%K A137974 nonn
%O A137974 0,3
%A A137974 _Paul D. Hanna_, Feb 26 2008