%I #20 Jan 13 2024 16:13:41
%S 0,0,0,0,2,0,0,18,6,0,0,192,72,48,0,0,2500,960,720,540,0,0,38880,
%T 15000,11520,9720,7680,0,0,705894,272160,210000,181440,161280,131250,
%U 0,0,14680064,5647152,4354560,3780000,3440640,3150000,2612736,0
%N Triangle read by rows: T(n, k) = binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k).
%C 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.
%H John Riordan and N. J. A. Sloane, <a href="http://dx.doi.org/10.1017/S1446788700007527">Enumeration of rooted trees by total height</a>, J. Austral. Math. Soc., vol. 10 pp. 278-282, 1969.
%e Triangle starts:
%e [0] [0]
%e [1] [0, 0]
%e [2] [0, 2, 0]
%e [3] [0, 18, 6, 0]
%e [4] [0, 192, 72, 48, 0]
%e [5] [0, 2500, 960, 720, 540, 0]
%e [6] [0, 38880, 15000, 11520, 9720, 7680, 0]
%e [7] [0, 705894, 272160, 210000, 181440, 161280, 131250, 0]
%e [8] [0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0]
%t A368849[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k);
%t Table[A368849[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Jan 13 2024 *)
%o (SageMath)
%o def T(n, k):
%o return binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k)
%o for n in range(0, 9): print([n], [T(n, k) for k in range(n + 1)])
%Y T(n, 1) = A066274(n) for n >= 1.
%Y T(n, 1)/(n - 1) = A000169(n) for n >= 2.
%Y T(n, n - 1) = 2*A081133(n) for n >= 1.
%Y Sum_{k=0..n} T(n, k) = A001864(n).
%Y (Sum_{k=0..n} T(n, k)) / n = A000435(n) for n >= 1.
%Y (Sum_{k=0..n} T(n, k)) * n / 2 = A262973(n) for n >= 1.
%Y (Sum_{k=2..n} T(n, k)) / (2*n) = A057500(n) for n >= 1.
%Y T(n, 1)/(n - 1) + (Sum_{k=2..n} T(n, k)) / (2*n) = A368951(n) for n >= 2.
%Y Sum_{k=0..n} (-1)^(k-1) * T(n, k) = A368981(n).
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Jan 11 2024