A137968 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^6.

1, 1, 6, 27, 158, 981, 6342, 42728, 295008, 2079882, 14908740, 108312873, 795836544, 5903472999, 44151306690, 332552305818, 2520416719368, 19207222744326, 147086508325056, 1131292622149352, 8735383810590486

Offset: 0

Author

Paul D. Hanna, Feb 26 2008

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A137967, A137969; A137970, A137972, A137974.

Programs

PARI

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^2)^6);polcoeff(A,n)}

PARI

a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(n-k),k)/(n-k)*binomial(2*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^6 where B(x) is the g.f. of A137967.

a(n) = Sum_{k=0..n-1} C(6*(n-k),k)/(n-k) * C(2*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009

a(n) ~ sqrt(6*s*(1-s)*(2-3*s) / ((44*s - 24)*Pi)) / (n^(3/2) * r^n), where r = 0.1201742080825038015263858974579392344239858277873... and s = 1.572098697306844482137442690518486437859864764710... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^6, 12 * r^2 * s * (1 + r*s^2)^5 = 1. - Vaclav Kotesovec, Nov 22 2017