A137957 G.f. satisfies A(x) = 1 + x*(1 + x*A(x)^4)^3.

1, 1, 3, 15, 79, 468, 2895, 18670, 123765, 838860, 5785503, 40473729, 286504086, 2048388112, 14770313397, 107290913232, 784380664232, 5766985753620, 42614014459911, 316304429143995, 2357275139670183, 17631888703154172

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

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

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

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

Mathematica

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 *)

Prog

(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)}
(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

Crossrefs

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

Sequence in context: A193913 A292097 A052755 * A002514 A319216 A093889

Adjacent sequences: A137954 A137955 A137956 * A137958 A137959 A137960

Keyword

nonn

Author

Paul D. Hanna, Feb 26 2008

Status

approved