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A137970
G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^3)^6.
6
1, 1, 6, 33, 236, 1776, 14148, 117070, 995568, 8653068, 76508562, 686035674, 6223653276, 57018806567, 526802616954, 4902775644477, 45919926029588, 432511043009679, 4094087001128088, 38927025591433926, 371607779425490280
OFFSET
0,3
LINKS
FORMULA
G.f.: A(x) = 1 + x*B(x)^6 where B(x) is the g.f. of A137969.
a(n) = Sum_{k=0..n-1} C(6*(n-k),k)/(n-k) * C(3*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009
a(n) ~ sqrt(6*s*(1-s)*(3-4*s) / ((102*s - 72)*Pi)) / (n^(3/2) * r^n), where r = 0.0971328555591006631243189792661187629516513365080... and s = 1.377827066365760014851094517875193622070040930150... are real roots of the system of equations s = 1 + r*(1 + r*s^3)^6, 18 * r^2 * s^2 * (1 + r*s^3)^5 = 1. - Vaclav Kotesovec, Nov 22 2017
PROG
(PARI) {a(n)=local(A=1+x*O(x^n)); for(i=0, n, A=1+x*(1+x*A^3)^6); polcoeff(A, n)}
(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, binomial(6*(n-k), k)/(n-k)*binomial(3*k, n-k-1))) \\ Paul D. Hanna, Jun 16 2009
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 26 2008
STATUS
approved