A137957 - OEIS

%I #10 Mar 03 2018 13:53:33

%S 1,1,3,15,79,468,2895,18670,123765,838860,5785503,40473729,286504086,

%T 2048388112,14770313397,107290913232,784380664232,5766985753620,

%U 42614014459911,316304429143995,2357275139670183,17631888703154172

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

%F G.f.: A(x) = 1 + x*B(x)^3 where B(x) is the g.f. of A137958.

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

%F a(n) ~ sqrt(3*s*(1-s)*(4-5*s) / ((88*s - 72)*Pi)) / (n^(3/2) * r^n), where r = 0.1243879037293364492255197677726812528516871521834... and s = 1.373172215091866448521512759142574301075022413158... are real roots of the system of equations s = 1 + r*(1 + r*s^4)^3, 12 * r^2 * s^3 * (1 + r*s^4)^2 = 1. - _Vaclav Kotesovec_, Nov 22 2017

%t Flatten[{1,Table[Sum[Binomial[3*(n-k),k]/(n-k)*Binomial[4*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^4)^3);polcoeff(A,n)}

%o (PARI) a(n)=if(n==0,1,sum(k=0,n-1,binomial(3*(n-k),k)/(n-k)*binomial(4*k,n-k-1))) \\ _Paul D. Hanna_, Jun 16 2009

%Y Cf. A137958, A137956; A137953, A137962, A137969.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 26 2008