A387403 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+3,k) * binomial(n+3,n-k), where i is the imaginary unit.

1, 8, 50, 280, 1484, 7616, 38304, 190080, 934560, 4564736, 22189024, 107476096, 519180480, 2502850560, 12046666752, 57912029184, 278136798720, 1334832967680, 6402435630080, 30695114813440, 147110418036736, 704860523102208, 3376580007936000, 16172904859238400

Offset

0,2

Links

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

Formula

n*(n+6)*a(n) = (n+3) * (2*(2*n+5)*a(n-1) + 4*(n+2)*a(n-2)) for n > 1.

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

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

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

Mathematica

Table[Sum[2^(n-k)*Binomial[n+3, n-2*k]*Binomial[2*k+3, k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vincenzo Librandi, Sep 04 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, 2^(n-k)*binomial(n+3, n-2*k)*binomial(2*k+3, k));
(Magma) [&+[2^(n-k) * Binomial(n+3, n-2*k) * Binomial(2*k+3, k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025

Crossrefs

Cf. A006139, A387401, A387402.

Cf. A374513.

Sequence in context: A030279 A133357 A395431 * A397117 A081675 A283277

Adjacent sequences: A387400 A387401 A387402 * A387404 A387405 A387406

Keyword

nonn

Author

Seiichi Manyama, Aug 29 2025

Status

approved