A386844 a(n) = Sum_{k=0..n} binomial(3*n+2,k) * binomial(3*n-k,n-k).

1, 8, 83, 942, 11177, 136164, 1688031, 21187546, 268409813, 3424751568, 43948343243, 566607282118, 7333422759873, 95225755205564, 1239995365588919, 16186010348814258, 211729232160358317, 2774813844884684712, 36425708310248816547, 478880147399497482142, 6304133921156502650777

Offset

0,2

Links

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

Formula

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

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

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

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

a(n) ~ 3^(3*n + 5/2) / (25 * sqrt(Pi*n) * 2^(n-1)). - Vaclav Kotesovec, Aug 07 2025

Mathematica

Table[Sum[Binomial[3*n+2, k]*Binomial[3*n-k, n-k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 06 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(3*n+2, k)*binomial(3*n-k, n-k));
(Magma) [&+[Binomial(3*n+2, k) * Binomial(3*n-k, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 06 2025

Crossrefs

Cf. A386843, A386845.

Sequence in context: A236671 A232226 A238811 * A307829 A039939 A199907

Adjacent sequences: A386841 A386842 A386843 * A386845 A386846 A386847

Keyword

nonn

Author

Seiichi Manyama, Aug 05 2025

Status

approved