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%I #10 Mar 03 2018 13:53:10
%S 1,1,3,21,136,1032,8139,66975,567417,4915386,43350639,387889254,
%T 3512655498,32133132074,296496163113,2756279003712,25790064341592,
%U 242699145598212,2295564345035100,21811226043019788,208084639385653938
%N G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^6)^3.
%H Vaclav Kotesovec, <a href="/A137969/b137969.txt">Table of n, a(n) for n = 0..350</a>
%F G.f.: A(x) = 1 + x*B(x)^3 where B(x) is the g.f. of A137970.
%F a(n) = Sum_{k=0..n-1} C(3*(n-k),k)/(n-k) * C(6*k,n-k-1) for n>0 with a(0)=1. - _Paul D. Hanna_, Jun 16 2009
%F a(n) ~ sqrt(3*s*(1-s)*(6-7*s) / ((204*s - 180)*Pi)) / (n^(3/2) * r^n), where r = 0.0971328555591006631243189792661187629516513365080... and s = 1.254068189138542668013320901661524162625316815207... are real roots of the system of equations s = 1 + r*(1 + r*s^6)^3, 18 * r^2 * s^5 * (1 + r*s^6)^2 = 1. - _Vaclav Kotesovec_, Nov 22 2017
%o (PARI) {a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^6)^3);polcoeff(A,n)}
%o (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(n-k),k)/(n-k)*binomial(6*k,n-k-1))) \\ _Paul D. Hanna_, Jun 16 2009
%Y Cf. A137970, A137968; A137953, A137957, A137962.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 26 2008
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