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A336182
a(n) = Sum_{k=0..n} (-3)^k * binomial(n,k)^3.
3
1, -2, -14, 136, 106, -8492, 35344, 395008, -4547462, -4838372, 365951356, -1601617712, -19715085584, 233866581856, 285409397056, -20406741254144, 90043530872218, 1169513126877676, -13961261999882204, -18779832792734384, 1270510266589738636, -5584024444211882792
OFFSET
0,2
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
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)
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 *)
PROG
(PARI) {a(n) = sum(k=0, n, (-3)^k*binomial(n, k)^3)}
CROSSREFS
Column k=3 of A336179.
Cf. A206180.
Sequence in context: A111424 A317356 A375507 * A224729 A355722 A303395
KEYWORD
sign
AUTHOR
Seiichi Manyama, Jul 10 2020
STATUS
approved