A137958 - OEIS

%I #10 Mar 03 2018 13:53:37

%S 1,1,4,18,100,587,3660,23640,157076,1066281,7363620,51568732,

%T 365369868,2614235293,18862816112,137096744232,1002785827620,

%U 7376023180645,54525165453672,404858512190316,3018190533410664,22581907465905018

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

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

%F 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

%F 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

%t 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 *)

%o (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)}

%o (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

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 26 2008