A336169 - OEIS

%I #20 Jul 10 2020 22:09:59

%S 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,

%T 719,112681,347279,20251,1,0,0,1,5039,7479361,166923119,58370761,

%U 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

%N 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).

%C 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.

%F G.f. of column k: Sum_{j>=0} (k*j)!/j!^k * x^j / (1+x)^(k*j+1).

%e Square array begins:

%e   1, 1, 1,      1,           1,               1, ...

%e   0, 0, 1,      5,          23,             119, ...

%e   1, 0, 1,     67,        2401,          112681, ...

%e   0, 0, 1,   1109,      347279,       166923119, ...

%e   1, 0, 1,  20251,    58370761,    302857024681, ...

%e   0, 0, 1, 391355, 10693893503, 616967236620839, ...

%t 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 *)

%Y Columns k=0-5 give: A059841, A000007, A000012, A124435, A336170, A336171.

%Y Rows n=0-1 give: A000012, A033312.

%Y Main diagonal gives A336172.

%Y Cf. A229142.

%K nonn,tabl

%O 0,12

%A _Seiichi Manyama_, Jul 10 2020