A137969 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^6)^3.

1, 1, 3, 21, 136, 1032, 8139, 66975, 567417, 4915386, 43350639, 387889254, 3512655498, 32133132074, 296496163113, 2756279003712, 25790064341592, 242699145598212, 2295564345035100, 21811226043019788, 208084639385653938

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Formula

G.f.: A(x) = 1 + x*B(x)^3 where B(x) is the g.f. of A137970.

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

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

Prog

(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)}
(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

Crossrefs

Cf. A137970, A137968; A137953, A137957, A137962.

Sequence in context: A141041 A079753 A346935 * A303349 A337467 A318041

Adjacent sequences: A137966 A137967 A137968 * A137970 A137971 A137972

Keyword

nonn

Author

Paul D. Hanna, Feb 26 2008

Status

approved