A278884 a(n) = binomial(3*n-1,n) * binomial(3*n,n)/(2*n+1).

1, 2, 30, 672, 18150, 546546, 17672928, 600935040, 21212454582, 770748371250, 28657235757150, 1085694550387200, 41778588391394400, 1628982594897249312, 64234570537702934400, 2557710564063135005184, 102714012593435476016982, 4155894894567674772785250, 169274181059121504574121550, 6935873114065443534326340000, 285716428631735196825345889350, 11826871503027977442890882704050, 491714173272153004121882711232000

Offset: 0

Author

Paul D. Hanna, Nov 29 2016

Keywords

• nonn
• changed

Comments

Central terms of triangles A278881 and A278882; a(n) = A278881(2*n,n) for n>=0.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..606

Crossrefs

Cf. A278881, A278882.

Programs

Mathematica

Table[(Binomial[3n-1,n]Binomial[3n,n])/(2n+1),{n,0,50}] (* Harvey P. Dale, Mar 26 2023 *)

PARI

{a(n) = binomial(3*n-1,n) * binomial(3*n,n) / (2*n+1)}
for(n=0,20,print1(a(n),", "))

Formula

4*n^2*(2*n-1)*(2*n+1)*a(n) - 9*(3*n-1)^2*(3*n-2)^2*a(n-1) = 0. - R. J. Mathar, Dec 02 2016

From Stefano Spezia, Sep 04 2025: (Start)

G.f.: (1 + 2*hypergeom([1/3, 1/3, 2/3, 2/3], [1/2, 1, 3/2],9^3*x/2^4])/3.

a(n) ~ 4^(-2*n-1)*9^(3*n)/(n^2*Pi). (End)