A351339 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is Sum_{j=0..n} k^j * j^(n-j).
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 18, 9, 0, 1, 5, 20, 48, 58, 23, 0, 1, 6, 30, 100, 201, 202, 66, 0, 1, 7, 42, 180, 516, 885, 762, 210, 0, 1, 8, 56, 294, 1105, 2756, 4116, 3114, 733, 0, 1, 9, 72, 448, 2094, 6955, 15300, 20298, 13754, 2781, 0
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, 6, ... 0, 2, 6, 12, 20, 30, 42, ... 0, 4, 18, 48, 100, 180, 294, ... 0, 9, 58, 201, 516, 1105, 2094, ... 0, 23, 202, 885, 2756, 6955, 15198, ... 0, 66, 762, 4116, 15300, 45030, 112686, ...
Programs
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Mathematica
T[0, k_] := 1; T[n_, 0] = 0; T[n_, k_] := Sum[k^j * j^(n - j), {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 08 2022 *) -
PARI
T(n, k) = sum(j=0, n, k^j*j^(n-j));
Formula
G.f. of column k: Sum_{j>=0} (k*x)^j/(1 - j*x).