A153293 G.f.: A(x) = F(x*F(x)^3) = F(F(x)-1) where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.

1, 1, 6, 42, 317, 2508, 20517, 172180, 1474689, 12843768, 113444721, 1014062898, 9158151426, 83449247979, 766340138037, 7085966319858, 65919413472834, 616559331247512, 5794778945023698, 54700034442193302, 518375457403431600

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

Cf. A001764, A000108; A153292, A153294.

Sequence in context: A034171 A264911 A244902 * A145301 A107266 A142985

Adjacent sequences: A153290 A153291 A153292 * A153294 A153295 A153296

Keyword

nonn

Author

Paul D. Hanna, Jan 14 2009

Status

approved