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

0,3

Links

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A084832 A135177 A244309 * A371543 A371723 A215522

Adjacent sequences: A137955 A137956 A137957 * A137959 A137960 A137961

Keyword

nonn

Author

Paul D. Hanna, Feb 26 2008

Status

approved