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A254747 a(n) = (1 + Sum_{j=0..n} (C(n,j)*C(3*j-1,j))) / 2.

%I A254747 #32 Oct 13 2017 03:30:44
%S A254747 1,2,8,47,312,2162,15311,109965,797824,5833298,42910998,317224800,
%T A254747 2354712927,17538747124,131017428431,981194304302,7364370502896,
%U A254747 55380344444150,417176211054422,3147365470080480,23777750075552262
%N A254747 a(n) = (1 + Sum_{j=0..n} (C(n,j)*C(3*j-1,j))) / 2.
%H A254747 G. C. Greubel, <a href="/A254747/b254747.txt">Table of n, a(n) for n = 0..1000</a>
%F A254747 G.f.: x*G'(x)/G(x), where G(x) = x*(2/sqrt(3*x*(1-x)))*sin((1/3)*asin(3/2*sqrt(3*x/(1-x)))).
%F A254747 a(n) = (hypergeom([1/3,2/3,-n],[1/2,1],-27/4)+2)/3. - _Peter Luschny_, Feb 07 2015
%F A254747 From _Vaclav Kotesovec_, Feb 07 2015: (Start)
%F A254747 Recurrence: 2*n*(2*n-1)*a(n) = (43*n^2 - 53*n + 18)*a(n-1) - 3*(35*n^2 - 85*n + 54)*a(n-2) + (n-2)*(97*n - 165)*a(n-3) - 31*(n-3)*(n-2)*a(n-4).
%F A254747 a(n) ~ 31^(n+1/2) / (9 * sqrt(Pi*n) * 2^(2*n+1)).
%F A254747 (End)
%p A254747 a := n -> (hypergeom([1/3, 2/3, -n], [1/2, 1], -27/4) +2 ) / 3:
%p A254747 seq(simplify(a(n)), n=0..20); # _Peter Luschny_, Feb 07 2015
%t A254747 FullSimplify[CoefficientList[Series[1 + x*D[Log[(2*Sin[(1/3)* ArcSin[(3/2)*Sqrt[3]* Sqrt[x/(1 - x)]]])/ (Sqrt[3]*Sqrt[(1 - x)* x])], x], {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Feb 07 2015 *)
%t A254747 Table[(1 + Sum[Binomial[n, j]*Binomial[3*j-1, j], {j, 0, n}])/2, {n,0,20}] (* _Vaclav Kotesovec_, Feb 07 2015 after _Vladimir Kruchinin_ *)
%o A254747 (Maxima)
%o A254747 a(n):=(1+sum(binomial(n,j)*binomial(3*j-1,j),j,0,n))/2;
%o A254747 (PARI) for(n=0,25, print1((1 + sum(k=0,n, binomial(n,k)*binomial(3*k-1,k)))/2, ", ")) \\ _G. C. Greubel_, Jun 03 2017
%Y A254747 Cf. A165817, A188687.
%K A254747 nonn
%O A254747 0,2
%A A254747 _Vladimir Kruchinin_, Feb 07 2015