A336182 a(n) = Sum_{k=0..n} (-3)^k * binomial(n,k)^3.

1, -2, -14, 136, 106, -8492, 35344, 395008, -4547462, -4838372, 365951356, -1601617712, -19715085584, 233866581856, 285409397056, -20406741254144, 90043530872218, 1169513126877676, -13961261999882204, -18779832792734384, 1270510266589738636, -5584024444211882792

Offset: 0

Author

Seiichi Manyama, Jul 10 2020

Keywords

• sign

Comments

Diagonal of the rational function 1 / (1 + y + z + x*y + y*z - 3*z*x - 2*x*y*z).

Diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) + 3*x*y*z).

Links

• Robert Israel, Table of n, a(n) for n = 0..1020

Crossrefs

Column k=3 of A336179.

Cf. A206180.

Programs

Maple

f:= gfun:-rectoproc({(24*n^3 + 176*n^2 + 416*n + 320)*a(n + 1) + (279*n^3 + 2325*n^2 + 6382*n + 5776)*a(n + 2) + (18*n^3 + 168*n^2 + 514*n + 512)*a(n + 3) + (3*n^3 + 31*n^2 + 104*n + 112)*a(n + 4), a(0) = 1, a(1) = -2, a(2) = -14, a(3) = 136},a(n),remember):
map(f, [$0..30]); # Robert Israel, Jul 12 2020

Mathematica

a[n_] := Sum[(-3)^k * Binomial[n, k]^3, {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Jul 11 2020 *)

PARI

{a(n) = sum(k=0, n, (-3)^k*binomial(n,k)^3)}

Formula

From Robert Israel, Jul 12 2020: (Start)

a(n) = hypergeom([-n,-n,-n],[1,1],3).

(24*n^3 + 176*n^2 + 416*n + 320)*a(n + 1) + (279*n^3 + 2325*n^2 + 6382*n + 5776)*a(n + 2) + (18*n^3 + 168*n^2 + 514*n + 512)*a(n + 3) + (3*n^3 + 31*n^2 + 104*n + 112)*a(n + 4)=0. (End)