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%I #9 Nov 02 2019 03:13:01
%S 1,1,6,42,317,2508,20517,172180,1474689,12843768,113444721,1014062898,
%T 9158151426,83449247979,766340138037,7085966319858,65919413472834,
%U 616559331247512,5794778945023698,54700034442193302,518375457403431600
%N 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.
%H Robert Israel, <a href="/A153293/b153293.txt">Table of n, a(n) for n = 0..994</a>
%F 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.
%F G.f. satisfies: A(x) = 1 + x*F(x)^3*A(x)^3 where F(x) is the g.f. of A001764.
%F 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.
%F 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
%e G.f.: A(x) = F(x*F(x)^3) = 1 + x + 6*x^2 + 42*x^3 + 317*x^4 +... where
%e F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
%e F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
%e F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
%p 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)):
%p 1, seq(simplify(S),n=1..40); # _Robert Israel_, Dec 26 2017
%t F[x_] = 1 + InverseSeries[x/(1 + x)^3 + O[x]^21];
%t CoefficientList[F[F[x] - 1], x] (* _Jean-François Alcover_, Nov 02 2019 *)
%o (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)))}
%Y Cf. A001764, A000108; A153292, A153294.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 14 2009
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