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A336179 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} (-k)^j * binomial(n,j)^3.

%I A336179 #26 Jul 12 2020 10:53:32
%S A336179 1,1,1,1,0,1,1,-1,-6,1,1,-2,-11,0,1,1,-3,-14,47,90,1,1,-4,-15,136,241,
%T A336179 0,1,1,-5,-14,261,106,-2281,-1680,1,1,-6,-11,416,-639,-8492,-3779,0,1,
%U A336179 1,-7,-6,595,-2294,-17523,35344,104831,34650,1,1,-8,1,792,-5135,-25624,188049,395008,-110207,0,1
%N A336179 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} (-k)^j * binomial(n,j)^3.
%C A336179 Column k is the diagonal of the rational function 1 / (1 + y + z + x*y + y*z - k*z*x - (k-1)*x*y*z).
%C A336179 Column k is the diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) + k*x*y*z).
%H A336179 Seiichi Manyama, <a href="/A336179/b336179.txt">Antidiagonals n = 0..139, flattened</a>
%e A336179 Square array begins:
%e A336179   1,  1,     1,     1,      1,      1, ...
%e A336179   1,  0,    -1,    -2,     -3,     -4, ...
%e A336179   1, -6,   -11,   -14,    -15,    -14, ...
%e A336179   1,  0,    47,   136,    261,    416, ...
%e A336179   1, 90,   241,   106,   -639,  -2294, ...
%e A336179   1,  0, -2281, -8492, -17523, -25624, ...
%t A336179 Unprotect[Power]; 0^0 = 1; T[n_, k_] := Sum[(-k)^j * Binomial[n, j]^3, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jul 11 2020 *)
%Y A336179 Columns k=0-3 give: A000012, A245086, A336181, A336182.
%Y A336179 Main diagonal gives A336180.
%Y A336179 Cf. A307884, A336163.
%K A336179 sign,tabl
%O A336179 0,9
%A A336179 _Seiichi Manyama_, Jul 10 2020