A388045 a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(3*n+1,k).

1, 9, 113, 1561, 22569, 335137, 5064793, 77500017, 1196924561, 18618616249, 291280485825, 4578268733705, 72239609939769, 1143593778271185, 18154539163583145, 288902926601894369, 4607221770289003041, 73610130453448492393, 1178030452561044788497, 18880762189892645812473

Offset

0,2

Links

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

Formula

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

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

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

a(n) = hypergeom([-1-3*n, -n], [1], 2). - Stefano Spezia, Sep 19 2025

a(n) = [x^n] (1+x)^(3*n+1) * (2+x)^n. - Seiichi Manyama, Sep 21 2025

Mathematica

Table[Sum[2^k*Binomial[n, k]*Binomial[3*n+1, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Sep 19 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 2^k*binomial(n, k)*binomial(3*n+1, k));
(Magma) [&+[2^k*Binomial(n, k)*Binomial(3*n+1, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 19 2025

Crossrefs

Column k=3 of A388052.

Cf. A002002, A108448, A388046.

Sequence in context: A241804 A156949 A155624 * A165224 A342296 A243676

Adjacent sequences: A388042 A388043 A388044 * A388046 A388047 A388048

Keyword

nonn

Author

Seiichi Manyama, Sep 14 2025

Status

approved