A379025 a(n) = Sum_{k=0..n} binomial(3*n+k-1,k) * binomial(3*n+k,n-k).

1, 6, 78, 1149, 17850, 285711, 4661727, 77086008, 1287322866, 21661521945, 366687839133, 6237631866417, 106535632157643, 1825763898882189, 31379978657609100, 540688387589377764, 9336602657251874754, 161534120354250452361, 2799488717098336992687

Offset: 0

Author

Seiichi Manyama, Dec 14 2024

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A262910, A379022, A379026.

Programs

Magma

[&+[Binomial(3*n+k-1, k)*Binomial(3*n+k, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Dec 21 2024

Mathematica

Table[Sum[Binomial[3*n+k-1,k]*Binomial[3*n+k, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Dec 21 2024 *)
a[n_]:= Binomial[3*n, n]*HypergeometricPFQ[{-n, 3*n, 1 + 3*n}, {1/2 + n, 1 + n}, -1/4]; Array[a,19,0] (* Stefano Spezia, Dec 22 2024 *)

PARI

a(n) = sum(k=0, n, binomial(3*n+k-1, k)*binomial(3*n+k, n-k));

Formula

a(n) = [x^n] ( (1 + x)/(1 - x - x^2) )^(3*n).

a(n) = binomial(3*n, n)*hypergeom([-n, 3*n, 1 + 3*n], [1/2 + n, 1 + n], -1/4). - Stefano Spezia, Dec 22 2024