A137960 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^2.

1, 1, 2, 11, 50, 275, 1560, 9212, 56082, 348675, 2207120, 14171155, 92075064, 604266000, 3999688050, 26670727220, 178997024610, 1208160130227, 8195828345756, 55849242272130, 382119958804520, 2624041637846210

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. A137961, A137959; A137952, A137955, A137967.

Programs

PARI

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^2);polcoeff(A,n)}

PARI

a(n)=if(n==0,1,sum(k=0,n-1,binomial(2*(n-k),k)/(n-k)*binomial(5*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^2 where B(x) is the g.f. of A137961.

a(n) = Sum_{k=0..n-1} C(2*(n-k),k)/(n-k) * C(5*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009

a(n) ~ sqrt(2*s*(1-s)*(5-6*s) / ((90*s - 80)*Pi)) / (n^(3/2) * r^n), where r = 0.1354712934479194768810666044866029126617104117352... and s = 1.354660923650925199331121807468321286698258863972... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^2, 10 * r^2 * s^4 * (1 + r*s^5) = 1. - Vaclav Kotesovec, Nov 22 2017