A137958 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^3)^4.

1, 1, 4, 18, 100, 587, 3660, 23640, 157076, 1066281, 7363620, 51568732, 365369868, 2614235293, 18862816112, 137096744232, 1002785827620, 7376023180645, 54525165453672, 404858512190316, 3018190533410664, 22581907465905018

Offset: 0

Author

Paul D. Hanna, Feb 26 2008

Keywords

• nonn

Crossrefs

Cf. A137957, A137959; A137956, A137964, A137971.

Programs

Mathematica

Flatten[{1,Table[Sum[Binomial[4*(n-k),k]/(n-k)*Binomial[3*k,n-k-1],{k,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, Nov 22 2017 *)

PARI

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=1+x*(1+x*A^3)^4);polcoeff(A,n)}

PARI

a(n)=if(n==0,1,sum(k=0,n-1,binomial(4*(n-k),k)/(n-k)*binomial(3*k,n-k-1))) \\ Paul D. Hanna, Jun 16 2009

Formula

G.f.: A(x) = 1 + x*B(x)^4 where B(x) is the g.f. of A137957.

a(n) = Sum_{k=0..n-1} C(4*(n-k),k)/(n-k) * C(3*k,n-k-1) for n>0 with a(0)=1. - Paul D. Hanna, Jun 16 2009

a(n) ~ sqrt(4*s*(1-s)*(3-4*s) / ((66*s - 48)*Pi)) / (n^(3/2) * r^n), where r = 0.1243879037293364492255197677726812528516871521834... and s = 1.442260525872978775674461288363175530136608288804... are real roots of the system of equations s = 1 + r*(1 + r*s^3)^4, 12 * r^2 * s^2 * (1 + r*s^3)^3 = 1. - Vaclav Kotesovec, Nov 22 2017