A387338 a(n) = Sum_{k=0..n} 2^k * binomial(n+3,k) * binomial(n+3,k+3).

1, 12, 100, 720, 4809, 30744, 191184, 1167120, 7033785, 41999364, 249075684, 1469561184, 8636441905, 50600529840, 295755641152, 1725379046496, 10050215851665, 58470232877820, 339832224226180, 1973538115293360, 11453616812552761, 66436765880135112

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Formula

n*(n+6)*a(n) = (n+3) * (3*(2*n+5)*a(n-1) - (n+2)*a(n-2)) for n > 1.

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

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

E.g.f.: exp(3*x) * BesselI(3, 2*sqrt(2)*x) / (2*sqrt(2)), with offset 3.

Mathematica

Table[Sum[2^k * Binomial[n+3, k]*Binomial[n+3, k+3], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 29 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 2^k*binomial(n+3, k)*binomial(n+3, k+3));
(Magma) [&+[2^k * Binomial(n+3, k) * Binomial(n+3, k+3): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025

Crossrefs

Cf. A002696, A387339.

Cf. A387342.

Sequence in context: A123902 A358302 A085374 * A387272 A016283 A243880

Adjacent sequences: A387335 A387336 A387337 * A387339 A387340 A387341

Keyword

nonn

Author

Seiichi Manyama, Aug 27 2025

Status

approved