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a(n) = floor(n^2 * (n - 1)^(n - 1) / 2).
+10
4
0, 0, 2, 18, 216, 3200, 56250, 1143072, 26353376, 679477248, 19371024450, 605000000000, 20542440283992, 753410487877632, 29681760446040794, 1250100767875276800, 56050417968750000000, 2665554518651030208512, 134012922425586555796674
MATHEMATICA
A369027[n_] := If[n == 1, 0, n^2 (n-1)^(n-1) / 2];
PROG
(SageMath)
def A369027(n): return ((n - 1)^(n - 1)*n^2)//2 print([ A369027(n) for n in range(22)])
a(n) = floor(n^(n - 1) / 2) for n > 0 and otherwise 0.
+10
3
0, 0, 1, 4, 32, 312, 3888, 58824, 1048576, 21523360, 500000000, 12968712300, 371504185344, 11649042561240, 396857386627072, 14596463012695312, 576460752303423488, 24330595937833434240, 1092955779869348265984
MATHEMATICA
A369026[n_] := If[n > 0, Floor[n^(n-1) / 2], 0];
PROG
(SageMath)
def A369026(n): return n^(n - 1) // 2 if n > 0 else 0 print([ A369026(n) for n in range(22)])
Triangle read by rows: T(n, k) = binomial(n, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k) / 2.
+10
2
0, 0, 0, 0, 1, 0, 0, 9, 3, 0, 0, 96, 36, 24, 0, 0, 1250, 480, 360, 270, 0, 0, 19440, 7500, 5760, 4860, 3840, 0, 0, 352947, 136080, 105000, 90720, 80640, 65625, 0, 0, 7340032, 2823576, 2177280, 1890000, 1720320, 1575000, 1306368, 0
EXAMPLE
Triangle starts:
[0] [0]
[1] [0, 0]
[2] [0, 1, 0]
[3] [0, 9, 3, 0]
[4] [0, 96, 36, 24, 0]
[5] [0, 1250, 480, 360, 270, 0]
[6] [0, 19440, 7500, 5760, 4860, 3840, 0]
[7] [0, 352947, 136080, 105000, 90720, 80640, 65625, 0]
[8] [0, 7340032, 2823576, 2177280, 1890000, 1720320, 1575000, 1306368, 0]
MAPLE
T := (n, k) -> binomial(n, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k)/2:
seq(seq(T(n, k), k = 0..n), n=0..9);
MATHEMATICA
A368982[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k)/2; Table[ A368982[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 28 2024 *)
PROG
(SageMath)
def T(n, k): return binomial(n, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k)//2
for n in range(0, 9): print([T(n, k) for k in range(n + 1)])
CROSSREFS
A368849, A369016 and this sequence are alternative sum representation for A001864 with different normalizations.
T(n, n - 1) = A081133(n - 2) for n >= 2. Sum_{k=0..n} T(n, k) = A036276(n - 1) for n >= 1. Sum_{k=0..n} (-1)^(k+1)*T(n, k) = A368981(n) / 2 for n >= 0.
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