OFFSET
0,5
COMMENTS
A motivation for this triangle was to provide an alternative sum representation for A001864(n) = n! * Sum_{k=0..n-2} n^k/k!. See formula 3 and formula 15 in Riordan and Sloane.
LINKS
John Riordan and N. J. A. Sloane, Enumeration of rooted trees by total height, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.
EXAMPLE
Triangle starts:
[0] [0]
[1] [0, 0]
[2] [0, 2, 0]
[3] [0, 18, 6, 0]
[4] [0, 192, 72, 48, 0]
[5] [0, 2500, 960, 720, 540, 0]
[6] [0, 38880, 15000, 11520, 9720, 7680, 0]
[7] [0, 705894, 272160, 210000, 181440, 161280, 131250, 0]
[8] [0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0]
MATHEMATICA
A368849[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k);
Table[A368849[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 13 2024 *)
PROG
(SageMath)
def T(n, k):
return binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k)
for n in range(0, 9): print([n], [T(n, k) for k in range(n + 1)])
CROSSREFS
T(n, 1) = A066274(n) for n >= 1.
T(n, 1)/(n - 1) = A000169(n) for n >= 2.
T(n, n - 1) = 2*A081133(n) for n >= 1.
Sum_{k=0..n} T(n, k) = A001864(n).
(Sum_{k=0..n} T(n, k)) / n = A000435(n) for n >= 1.
(Sum_{k=0..n} T(n, k)) * n / 2 = A262973(n) for n >= 1.
(Sum_{k=2..n} T(n, k)) / (2*n) = A057500(n) for n >= 1.
T(n, 1)/(n - 1) + (Sum_{k=2..n} T(n, k)) / (2*n) = A368951(n) for n >= 2.
Sum_{k=0..n} (-1)^(k-1) * T(n, k) = A368981(n).
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Jan 11 2024
STATUS
approved