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

1, 2, 8, 47, 312, 2162, 15311, 109965, 797824, 5833298, 42910998, 317224800, 2354712927, 17538747124, 131017428431, 981194304302, 7364370502896, 55380344444150, 417176211054422, 3147365470080480, 23777750075552262

Offset: 0

Author

Vladimir Kruchinin, Feb 07 2015

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A165817, A188687.

Programs

Maple

a := n -> (hypergeom([1/3, 2/3, -n], [1/2, 1], -27/4) +2 ) / 3:
seq(simplify(a(n)), n=0..20); # Peter Luschny, Feb 07 2015

Mathematica

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 *)
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 *)

Maxima

a(n):=(1+sum(binomial(n,j)*binomial(3*j-1,j),j,0,n))/2;

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

Formula

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)))).

a(n) = (hypergeom([1/3,2/3,-n],[1/2,1],-27/4)+2)/3. - Peter Luschny, Feb 07 2015

From Vaclav Kotesovec, Feb 07 2015: (Start)

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).

a(n) ~ 31^(n+1/2) / (9 * sqrt(Pi*n) * 2^(2*n+1)).

(End)