A387339 a(n) = Sum_{k=0..n} 3^k * binomial(n+2,k) * binomial(n+2,k+2).

1, 12, 108, 880, 6855, 52164, 391720, 2918304, 21634290, 159880600, 1179180552, 8685874080, 63930198787, 470327654580, 3459353475600, 25442360389696, 187126561024686, 1376455855989672, 10126540146288520, 74515694338112160, 548444877468906726

Offset

0,2

Links

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

Formula

n*(n+4)*a(n) = (n+2) * (4*(2*n+3)*a(n-1) - 4*(n+1)*a(n-2)) for n > 1.

a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).

a(n) = [x^n] (1+4*x+3*x^2)^(n+2).

E.g.f.: exp(4*x) * BesselI(2, 2*sqrt(3)*x) / 3, with offset 2.

Mathematica

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

Prog

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

Crossrefs

Cf. A069835, A331792, A387340.

Cf. A387310.

Sequence in context: A037972 A111990 A053469 * A055533 A037602 A037707

Adjacent sequences: A387336 A387337 A387338 * A387340 A387341 A387342

Keyword

nonn

Author

Seiichi Manyama, Aug 27 2025

Status

approved