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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).
3
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
0,12
COMMENTS
Column k is the diagonal of the rational function 1 / (1 - Sum_{j=1..k} x_j + Product_{j=1..k} x_j) for k>0.
FORMULA
G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1+x)^(k*j+1).
EXAMPLE
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, ...
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 *)
CROSSREFS
Columns k=0-5 give: A059841, A000007, A000012, A124435, A336170, A336171.
Rows n=0-1 give: A000012, A033312.
Main diagonal gives A336172.
Cf. A229142.
Sequence in context: A293087 A209349 A242095 * A007912 A019755 A085475
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jul 10 2020
STATUS
approved