A336201 - OEIS

%I #18 May 01 2021 17:40:09

%S 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,

%T 1,-5,-134,909,106,-81,0,1,1,-6,-347,4736,3585,-8492,141,0,1,1,-7,

%U -846,21655,61906,-323523,35344,363,0,1,1,-8,-1983,91512,771601,-8065624,2201809,395008,-1791,0,1

%N 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.

%C 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.

%e Square array begins:

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

%e   1, 0,  -1,    -2,      -3,       -4, ...

%e   1, 0,  -3,   -14,     -47,     -134, ...

%e   1, 0,  11,   136,     909,     4736, ...

%e   1, 0,   1,   106,    3585,    61906, ...

%e   1, 0, -81, -8492, -323523, -8065624, ...

%t 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 *)

%Y Columns k=0-3 give: A000012, A000007, (-1)^n*A098332(n), A336182.

%Y Main diagonal gives A336202.

%Y Cf. A309010, A336179, A336187.

%K sign,tabl

%O 0,12

%A _Seiichi Manyama_, Jul 11 2020