A387401 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+1,k) * binomial(n+1,n-k), where i is the imaginary unit.

1, 4, 18, 80, 360, 1632, 7448, 34176, 157536, 728960, 3384128, 15754752, 73525504, 343870464, 1611288960, 7562801152, 35550504448, 167339022336, 788643765248, 3720901222400, 17573439614976, 83074892775424, 393056192851968, 1861155016212480, 8819174122700800, 41818448615636992

Offset

0,2

Links

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

Formula

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

a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).

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

E.g.f.: exp(2*x) * BesselI(1, 2*sqrt(2)*x) / sqrt(2), with offset 1.

a(n) = (n+1) * A071356(n).

Mathematica

Table[Sum[2^(n-k)*Binomial[n+1, n-2*k]*Binomial[2*k+1, 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+1, n-2*k)*binomial(2*k+1, k));
(Magma) [&+[2^(n-k) * Binomial(n+1, n-2*k) * Binomial(2*k+1, k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025

Crossrefs

Cf. A006139, A387402, A387403.

Cf. A071356, A374497.

Sequence in context: A045902 A090017 A257390 * A104631 A106391 A063881

Adjacent sequences: A387398 A387399 A387400 * A387402 A387403 A387404

Keyword

nonn

Author

Seiichi Manyama, Aug 29 2025

Status

approved