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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).
3
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
0,8
FORMULA
T = B066320 - A369017 (where B066320 = A066320 after adding a 0-column to the left and then setting offset to (0, 0)).
EXAMPLE
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]
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);
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 *)
PROG
(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)])
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.
Sequence in context: A224476 A123147 A119831 * A224842 A318138 A130143
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 12 2024
STATUS
approved