OFFSET
0,3
LINKS
Robert Israel, Table of n, a(n) for n = 0..994
FORMULA
a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(3n,n-k)*k/n for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*F(x)^3*A(x)^3 where F(x) is the g.f. of A001764.
G.f. satisfies: A(x/G(x)) = F(x*G(x)^2) = F(G(x)-1) where G(x) = F(x/G(x)) is the g.f. of A000108 and F(x) is the g.f. of A001764.
a(n) = sqrt(3)*Gamma(n+2/3)*Gamma(n+1/3)*hypergeom([4/3, 5/3, -n+1], [5/2, 2*n+2], -27/4)*27^n/(2*Pi*(n+1)!) for n >= 1. - Robert Israel, Dec 26 2017
EXAMPLE
G.f.: A(x) = F(x*F(x)^3) = 1 + x + 6*x^2 + 42*x^3 + 317*x^4 +... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
MAPLE
S:= (1/2)*GAMMA(n+1/3)*GAMMA(n+2/3)*hypergeom([4/3, 5/3, -n+1], [5/2, 2*n+2], -27/4)*27^n*sqrt(3)/(Pi*GAMMA(2*n+2)):
1, seq(simplify(S), n=1..40); # Robert Israel, Dec 26 2017
MATHEMATICA
F[x_] = 1 + InverseSeries[x/(1 + x)^3 + O[x]^21];
CoefficientList[F[F[x] - 1], x] (* Jean-François Alcover, Nov 02 2019 *)
PROG
(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(3*k+1, k)/(3*k+1)*binomial(3*(n-k)+3*k, n-k)*3*k/(3*(n-k)+3*k)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 14 2009
STATUS
approved