A387337 a(n) = Sum_{k=0..n} 2^k * binomial(n+2,k) * binomial(n+2,k+2).

1, 9, 62, 390, 2355, 13923, 81340, 471852, 2725365, 15698925, 90278298, 518620050, 2977523367, 17089401735, 98073111800, 562837648728, 3230477449641, 18545060127249, 106484493098230, 611580138259230, 3513476242850651, 20190255588746859, 116056618927038132

Offset: 0

Author

Seiichi Manyama, Aug 27 2025

Keywords

• nonn
• new

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A050151.

Programs

Magma

[&+[2^k * Binomial(n+2,k) * Binomial(n+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025

Mathematica

Table[Sum[2^k*Binomial[n+2, k] * Binomial[n+2, k+2], {k,0,n}], {n,0,25}] (* Vaclav Kotesovec, Aug 27 2025 *)

PARI

a(n) = sum(k=0, n, 2^k*binomial(n+2, k)*binomial(n+2, k+2));

Formula

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

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

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

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

a(n) ~ (1 + sqrt(2))^(2*n+5) / (2^(9/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 27 2025