A387276 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n+3,k+3) * binomial(2*k+6,k+6).

1, 20, 255, 2650, 24521, 210840, 1725234, 13631700, 104993955, 793367300, 5907885412, 43495473840, 317355930255, 2298888740400, 16555878011448, 118661449810320, 847132614218907, 6027874235210700, 42773816956415055, 302816249208061050, 2139537520524710691

Offset

0,2

Links

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

Formula

n*(n+6)*a(n) = (n+3) * (5*(2*n+5)*a(n-1) - 21*(n+2)*a(n-2)) for n > 1.

a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).

a(n) = [x^n] (1+5*x+x^2)^(n+3).

E.g.f.: exp(5*x) * BesselI(3, 2*x), with offset 3.

Mathematica

Table[Sum[3^(n-k)*Binomial[n+3, k+3]*Binomial[2*k+6, k+6], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Aug 31 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n+3, k+3)*binomial(2*k+6, k+6));
(Magma) [&+[3^(n-k) * Binomial(n+3, k+3) * Binomial(2*k+6, k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025

Crossrefs

Cf. A387275, A387277, A387278.

Cf. A387239, A387273.

Sequence in context: A241228 A306290 A075469 * A028032 A348055 A025986

Adjacent sequences: A387273 A387274 A387275 * A387277 A387278 A387279

Keyword

nonn

Author

Seiichi Manyama, Aug 24 2025

Status

approved