A137974 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^6.

1, 1, 6, 45, 410, 4020, 41826, 452207, 5033910, 57300285, 663912420, 7804131660, 92838682242, 1115595461915, 13521340799310, 165104951405235, 2029162664033790, 25081468301798301, 311593507408597920

Offset: 0

Author

Paul D. Hanna, Feb 26 2008

Keywords

• nonn

Comments

In general, if g.f. satisfies A(x) = 1 + x*(1 + x*A(x)^p)^q, p >= 1, q >= 1, p + q > 2, then a(n) ~ sqrt(q*s*(1-s)*(p*(1-s)-s) / (2*Pi*p*(q-s-p*q*(1-s)))) / (n^(3/2) * r^n), where r and s are real roots of the system of equations s = 1 + r*(1 + r*s^p)^q, p*q * r^2 * s^(p-1) * (1 + r*s^p)^(q-1) = 1. - Vaclav Kotesovec, Nov 22 2017

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A137973; A137968, A137970, A137972.

Programs

PARI

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^6);polcoeff(A,n)}

PARI

a(n)=if(n==0,1,sum(k=0,n-1,binomial(6*(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)^6 where B(x) is the g.f. of A137973.

a(n) = Sum_{k=0..n-1} C(6*(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(6*s*(1-s)*(5-6*s) / ((290*s - 240)*Pi)) / (n^(3/2) * r^n), where r = 0.0739607593319208338998816978154858830062403258604... and s = 1.234938729532398384561936758596402363403570701060... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^6, 30 * r^2 * s^4 * (1 + r*s^5)^5 = 1. - Vaclav Kotesovec, Nov 22 2017