A348618 a(n) = (1+(-1)^n)/2*4^n*(C((3*n)/2-1,n))+(1-(-1)^n)/2*((C((3*n-1)/2,n))*(C(3*n-1,(3*n-1)/2)))/(C(n-1,(n-1)/2)).

1, 2, 16, 140, 1280, 12012, 114688, 1108536, 10813440, 106234700, 1049624576, 10418726760, 103817412608, 1037865473400, 10404558274560, 104557533120240, 1052941297385472, 10623352887172620, 107358720517734400, 1086563988284497800, 11011614449734778880

Offset: 0

Author

Vladimir Kruchinin, Oct 25 2021

Keywords

• nonn

Crossrefs

Cf. A244038.

Programs

Maple

a:= n-> ceil(4^n*binomial(3*n/2, n)/3):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 25 2021

Mathematica

a[n_] := If[EvenQ[n], 4^n * Binomial[3*n/2 - 1, n], Binomial[(3*n - 1)/2, n] * Binomial[3*n - 1, (3*n - 1)/2] / Binomial[n - 1, (n - 1)/2]]; Array[a, 18, 0] (* Amiram Eldar, Oct 25 2021 *)

Maxima

a(n):=if evenp(n) then 4^n*binomial(3*n/2-1,n) else ((binomial((3*n-1)/2,n))*
    (binomial(3*n-1,(3*n-1)/2)))/binomial(n-1,(n-1)/2);

Formula

G.f.: (288*x^2*cos(arcsin(216*x^2-1)/3))/(sqrt(432*x^2-46656*x^4)*(2*sin(arcsin(216*x^2-1)/3)+1)).

Conjecture: D-finite with recurrence n*(n-1)*a(n) -12*(3*n-2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Mar 06 2022