A137961 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^2)^5.

1, 1, 5, 20, 105, 575, 3306, 19760, 121035, 757230, 4815530, 31039402, 202335855, 1331569725, 8834918160, 59035907480, 396937508816, 2683518356850, 18230570856710, 124390574548960, 852074529347120, 5857453791938135

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. A137960, A137962; A137963, A137965, A137973.

Programs

PARI

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

PARI

a(n)=if(n==0,1,sum(k=0,n-1,binomial(5*(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)^5 where B(x) is the g.f. of A137960.

a(n) = Sum_{k=0..n-1} C(5*(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(5*s*(1-s)*(2-3*s) / ((36*s - 20)*Pi)) / (n^(3/2) * r^n), where r = 0.1354712934479194768810666044866029126617104117352... and s = 1.618016818966151244027202981456410137451426090894... are real roots of the system of equations s = 1 + r*(1 + r*s^2)^5, 10 * r^2 * s * (1 + r*s^2)^4 = 1. - Vaclav Kotesovec, Nov 22 2017