A137962 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^5)^3.

1, 1, 3, 18, 106, 720, 5085, 37493, 284331, 2204973, 17404720, 139369905, 1129411314, 9244823986, 76326154857, 634847759955, 5314684735045, 44746683774474, 378652035541761, 3218705637379698, 27471657413667780

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. A137963, A137961; A137953, A137957, A137969.

Programs

PARI

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^5)^3);polcoeff(A,n)}

PARI

a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(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)^3 where B(x) is the g.f. of A137963.

a(n) = Sum_{k=0..n-1} C(3*(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(3*s*(1-s)*(5-6*s) / ((140*s - 120)*Pi)) / (n^(3/2) * r^n), where r = 0.1085884782751570249717333800652227343328635496829... and s = 1.301018963559115613510052458264916439485131890857... are real roots of the system of equations s = 1 + r*(1 + r*s^5)^3, 15 * r^2 * s^4 * (1 + r*s^5)^2 = 1. - Vaclav Kotesovec, Nov 22 2017