A336169 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * multinomial(n+(k-1)*j; n-j, {j}^k).
1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 5, 1, 0, 1, 1, 23, 67, 1, 0, 0, 1, 119, 2401, 1109, 1, 0, 1, 1, 719, 112681, 347279, 20251, 1, 0, 0, 1, 5039, 7479361, 166923119, 58370761, 391355, 1, 0, 1, 1, 40319, 681040081, 137127810959, 302857024681, 10693893503, 7847155, 1, 0, 0, 1, 362879, 81729285121, 182499151015439, 3244063941457921, 616967236620839, 2071837562929, 161476565, 1, 0, 1
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, ... 0, 0, 1, 5, 23, 119, ... 1, 0, 1, 67, 2401, 112681, ... 0, 0, 1, 1109, 347279, 166923119, ... 1, 0, 1, 20251, 58370761, 302857024681, ... 0, 0, 1, 391355, 10693893503, 616967236620839, ...
Crossrefs
Programs
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Mathematica
T[n_, k_] := Sum[(-1)^(n - j)*(n + (k - 1)*j)!/(n - j)!/(j!)^k, {j, 0, n} ]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 10 2020 *)
Formula
G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1+x)^(k*j+1).
Comments