A385667 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).

1, 10, 151, 2542, 44983, 819160, 15197404, 285653350, 5421341311, 103659081034, 1993769491591, 38532753357064, 747680491747876, 14556620712375856, 284217498703106224, 5563106991308471062, 109124768598722692111, 2144648671343440349182

Offset: 0

Author

Seiichi Manyama, Aug 04 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..350

Crossrefs

Cf. A384950.

Programs

Magma

[&+[2^(n-k) * Binomial(3*n+1,k) * Binomial(3*n-k,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 05 2025

Mathematica

Table[Sum[2^(n-k)*Binomial[3*n+1,k]*Binomial[3*n-k,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 05 2025 *)

PARI

a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+1, k)*binomial(3*n-k, n-k));

Formula

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

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

a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n+1,k).

a(n) = Sum_{k=0..n} 3^k * binomial(2*n+k,k).