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
Keywords
Examples
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 +...
Links
- Robert Israel, Table of n, a(n) for n = 0..994
Programs
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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
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Mathematica
F[x_] = 1 + InverseSeries[x/(1 + x)^3 + O[x]^21]; CoefficientList[F[F[x] - 1], x] (* Jean-François Alcover, Nov 02 2019 *)
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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)))}
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