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A336182
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a(n) = Sum_{k=0..n} (-3)^k * binomial(n,k)^3.
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3
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1, -2, -14, 136, 106, -8492, 35344, 395008, -4547462, -4838372, 365951356, -1601617712, -19715085584, 233866581856, 285409397056, -20406741254144, 90043530872218, 1169513126877676, -13961261999882204, -18779832792734384, 1270510266589738636, -5584024444211882792
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OFFSET
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0,2
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COMMENTS
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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).
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LINKS
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Robert Israel, Table of n, a(n) for n = 0..1020
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FORMULA
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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)
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MAPLE
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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
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MATHEMATICA
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a[n_] := Sum[(-3)^k * Binomial[n, k]^3, {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Jul 11 2020 *)
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PROG
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(PARI) {a(n) = sum(k=0, n, (-3)^k*binomial(n, k)^3)}
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CROSSREFS
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Column k=3 of A336179.
Cf. A206180.
Sequence in context: A326886 A111424 A317356 * A224729 A355722 A303395
Adjacent sequences: A336179 A336180 A336181 * A336183 A336184 A336185
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KEYWORD
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sign
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AUTHOR
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Seiichi Manyama, Jul 10 2020
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STATUS
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approved
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