A137965 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^4)^5.

1, 1, 5, 30, 220, 1725, 14356, 124020, 1102770, 10023680, 92722620, 870039474, 8261024380, 79225392830, 766302511445, 7466883915800, 73227699088806, 722228333119200, 7159117292177840, 71284856957207030, 712673042497177450

Offset: 0

Author

Paul D. Hanna, Feb 26 2008

Keywords

• nonn

Links

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

Crossrefs

Cf. A137964, A137966; A137961, A137963, A137973.

Programs

PARI

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^4)^5);polcoeff(A,n)}

PARI

a(n)=if(n==0,1,sum(k=0,n-1,binomial(5*(n-k),k)/(n-k)*binomial(4*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 A137964.

a(n) = Sum_{k=0..n-1} C(5*(n-k),k)/(n-k) * C(4*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009

a(n) ~ sqrt(5*s*(1-s)*(4-5*s) / ((152*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.0927175295193852172913829423030505161354091369581... and s = 1.306924048536121339538817141295744998528778296640... are real roots of the system of equations s = 1 + r*(1 + r*s^4)^5, 20 * r^2 * s^3 * (1 + r*s^4)^4 = 1. - Vaclav Kotesovec, Nov 22 2017