A369027
a(n) = floor(n^2 * (n - 1)^(n - 1) / 2).
Original entry on oeis.org
0, 0, 2, 18, 216, 3200, 56250, 1143072, 26353376, 679477248, 19371024450, 605000000000, 20542440283992, 753410487877632, 29681760446040794, 1250100767875276800, 56050417968750000000, 2665554518651030208512, 134012922425586555796674
Offset: 0
A369026
a(n) = floor(n^(n - 1) / 2) for n > 0 and otherwise 0.
Original entry on oeis.org
0, 0, 1, 4, 32, 312, 3888, 58824, 1048576, 21523360, 500000000, 12968712300, 371504185344, 11649042561240, 396857386627072, 14596463012695312, 576460752303423488, 24330595937833434240, 1092955779869348265984
Offset: 0
A368982
Triangle read by rows: T(n, k) = binomial(n, k - 1) * (k - 1)^(k - 1) * (n - k) * (n - k + 1)^(n - k) / 2.
Original entry on oeis.org
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
Offset: 0
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]
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.
-
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);
-
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 *)
-
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)])
Showing 1-3 of 3 results.