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A336201
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)^k.
1
1, 1, 1, 1, 0, 1, 1, -1, 0, 1, 1, -2, -3, 0, 1, 1, -3, -14, 11, 0, 1, 1, -4, -47, 136, 1, 0, 1, 1, -5, -134, 909, 106, -81, 0, 1, 1, -6, -347, 4736, 3585, -8492, 141, 0, 1, 1, -7, -846, 21655, 61906, -323523, 35344, 363, 0, 1, 1, -8, -1983, 91512, 771601, -8065624, 2201809, 395008, -1791, 0, 1
OFFSET
0,12
COMMENTS
Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) + k * Product_{j=1..k} x_j) for k>0.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 0, -1, -2, -3, -4, ...
1, 0, -3, -14, -47, -134, ...
1, 0, 11, 136, 909, 4736, ...
1, 0, 1, 106, 3585, 61906, ...
1, 0, -81, -8492, -323523, -8065624, ...
MATHEMATICA
T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j]^k, {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)
CROSSREFS
Columns k=0-3 give: A000012, A000007, (-1)^n*A098332(n), A336182.
Main diagonal gives A336202.
Sequence in context: A243081 A287847 A377666 * A271369 A308322 A308898
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Jul 11 2020
STATUS
approved