A369019 Triangle read by rows: T(n, k) = binomial(n, k - 1)*(k - 1)^(k - 1)*k*(n - k + 1)^(n - k).
0, 0, 1, 0, 2, 4, 0, 9, 12, 36, 0, 64, 72, 144, 432, 0, 625, 640, 1080, 2160, 6400, 0, 7776, 7500, 11520, 19440, 38400, 112500, 0, 117649, 108864, 157500, 241920, 403200, 787500, 2286144, 0, 2097152, 1882384, 2612736, 3780000, 5734400, 9450000, 18289152, 52706752
Offset: 0
Examples
Triangle starts: [0] [0] [1] [0, 1] [2] [0, 2, 4] [3] [0, 9, 12, 36] [4] [0, 64, 72, 144, 432] [5] [0, 625, 640, 1080, 2160, 6400] [6] [0, 7776, 7500, 11520, 19440, 38400, 112500] [7] [0, 117649, 108864, 157500, 241920, 403200, 787500, 2286144]
Programs
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Maple
T := (n, k) -> binomial(n, k - 1)*(k - 1)^(k - 1)*k*(n - k + 1)^(n - k): seq(seq(T(n, k), k = 0..n), n=0..9);
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Mathematica
A369019[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] k (n-k+1)^(n-k); Table[A369019[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 27 2024 *)
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SageMath
def A369019(n, k): return binomial(n, k - 1)*(k - 1)^(k - 1)*k*(n - k + 1)^(n - k)