A388134 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n,k) * binomial(3*n+1,k).

1, 7, 72, 822, 9844, 121191, 1519152, 19286364, 247171596, 3190893132, 41432815200, 540540451710, 7079710744764, 93033644833330, 1226008203218880, 16196061588290808, 214414233744532476, 2843907325928278644, 37783572774935248608, 502735026400871414136, 6698202844579014342096

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..882

Formula

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

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

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

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

Mathematica

Table[Sum[3^(n-k)* Binomial[n, k]*Binomial[3*n+1, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 24 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n, k)*binomial(3*n+1, k));
(Magma) [&+[3^(n-k)*Binomial(n, k)*Binomial(3*n+1, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 24 2025

Crossrefs

Cf. A098663, A388133, A388135.

Cf. A387931, A388049.

Sequence in context: A242827 A077332 A222833 * A005413 A306377 A394781

Adjacent sequences: A388131 A388132 A388133 * A388135 A388136 A388137

Keyword

nonn

Author

Seiichi Manyama, Sep 15 2025

Status

approved