A333991 a(n) = Sum_{k=0..n} (-n)^k * binomial(2*n,2*k).

1, 0, -7, 64, -527, 3776, -7199, -712704, 28545857, -881543168, 25615822601, -733594255360, 20859188600881, -580152163418112, 15048530008948913, -311489672222081024, 713562283940993281, 511135051171610230784, -48010258775057340355559, 3439412411849176925601792

Offset: 0

Author

Seiichi Manyama, Sep 04 2020

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..386

Crossrefs

Main diagonal of A333989.

Cf. A065440, A333990.

Programs

Mathematica

a[0] = 1; a[n_] := Sum[(-n)^k * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Sep 04 2020 *)
Table[Hypergeometric2F1[1/2 - n, -n, 1/2, -n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 05 2020 *)
Table[Cos[2*n*ArcTan[Sqrt[n]]] * (n + 1)^n, {n, 0, 20}] // Round (* Vaclav Kotesovec, Sep 05 2020 *)

PARI

{a(n) = sum(k=0, n, (-n)^k*binomial(2*n, 2*k))}

Formula

From Vaclav Kotesovec, Sep 05 2020: (Start)

a(n) = hypergeometric2F1(1/2 - n, -n, 1/2, -n).

a(n) = (1 + i*sqrt(n))^(2*n)/2 + (1 - i*sqrt(n))^(2*n)/2, where i is the imaginary unit.

a(n) = cos(2*n*arctan(sqrt(n))) * (n + 1)^n. (End)