A369016 Triangle read by rows: T(n, k) = binomial(n - 1, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k - 1).
0, 0, 0, 0, 1, 0, 0, 6, 2, 0, 0, 48, 18, 12, 0, 0, 500, 192, 144, 108, 0, 0, 6480, 2500, 1920, 1620, 1280, 0, 0, 100842, 38880, 30000, 25920, 23040, 18750, 0, 0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0
Offset: 0
Examples
Triangle starts: [0] [0] [1] [0, 0] [2] [0, 1, 0] [3] [0, 6, 2, 0] [4] [0, 48, 18, 12, 0] [5] [0, 500, 192, 144, 108, 0] [6] [0, 6480, 2500, 1920, 1620, 1280, 0] [7] [0, 100842, 38880, 30000, 25920, 23040, 18750, 0] [8] [0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0]
Crossrefs
A368849, A368982 and this sequence are alternative sum representation for A001864 with different normalizations.
T(n, k) = A368849(n, k) / n for n >= 1.
T(n, 1) = A053506(n) for n >= 1.
T(n, n - 1) = A055897(n - 1) for n >= 2.
Sum_{k=0..n} T(n, k) = A000435(n) for n >= 1.
Sum_{k=0..n} (-1)^(k+1)*T(n, k) = A368981(n) / n for n >= 1.
Programs
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Maple
T := (n, k) -> binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1): seq(seq(T(n, k), k = 0..n), n=0..9);
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Mathematica
A369016[n_, k_] := Binomial[n-1, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k-1); Table[A369016[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 28 2024 *)
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SageMath
def T(n, k): return binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1) for n in range(0, 9): print([T(n, k) for k in range(n + 1)])