A387402 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+2,k) * binomial(n+2,n-k), where i is the imaginary unit.

1, 6, 32, 160, 780, 3752, 17920, 85248, 404640, 1918400, 9090048, 43064320, 204032192, 966887040, 4583424000, 21735350272, 103114538496, 489392157696, 2323701678080, 11037970513920, 52454251902976, 249373626208256, 1186024281341952, 5642924625100800, 26858183388774400, 127880625111662592

Offset

0,2

Links

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

Formula

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

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

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

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

Mathematica

Table[Sum[2^(n-k)*Binomial[n+2, n-2*k]*Binomial[2*k+2, 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+2, n-2*k)*binomial(2*k+2, k));
(Magma) [&+[2^(n-k) * Binomial(n+2, n-2*k) * Binomial(2*k+2, k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025

Crossrefs

Cf. A006139, A387401, A387403.

Cf. A374511.

Sequence in context: A232331 A231992 A292044 * A006668 A232494 A037530

Adjacent sequences: A387399 A387400 A387401 * A387403 A387404 A387405

Keyword

nonn

Author

Seiichi Manyama, Aug 29 2025

Status

approved