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A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k).

%I A368849 #23 Apr 26 2025 20:53:34
%S A368849 0,0,0,0,2,0,0,18,6,0,0,192,72,48,0,0,2500,960,720,540,0,0,38880,
%T A368849 15000,11520,9720,7680,0,0,705894,272160,210000,181440,161280,131250,
%U A368849 0,0,14680064,5647152,4354560,3780000,3440640,3150000,2612736,0
%N A368849 Triangle read by rows: T(n, k) = binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k).
%C A368849 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 A368849 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 A368849 Triangle starts:
%e A368849   [0] [0]
%e A368849   [1] [0,        0]
%e A368849   [2] [0,        2,       0]
%e A368849   [3] [0,       18,       6,       0]
%e A368849   [4] [0,      192,      72,      48,      0]
%e A368849   [5] [0,     2500,     960,     720,     540,       0]
%e A368849   [6] [0,    38880,   15000,   11520,    9720,    7680,       0]
%e A368849   [7] [0,   705894,  272160,  210000,  181440,  161280,  131250,       0]
%e A368849   [8] [0, 14680064, 5647152, 4354560, 3780000, 3440640, 3150000, 2612736, 0]
%t A368849 A368849[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] (n-k) (n-k+1)^(n-k);
%t A368849 Table[A368849[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Jan 13 2024 *)
%o A368849 (SageMath)
%o A368849 def T(n, k):
%o A368849     return binomial(n, k - 1)*(k - 1)^(k - 1)*(n - k)*(n - k + 1)^(n - k)
%o A368849 for n in range(0, 9): print([n], [T(n, k) for k in range(n + 1)])
%Y A368849 T(n, 1) = A066274(n) for n >= 1.
%Y A368849 T(n, 1)/(n - 1) = A000169(n) for n >= 2.
%Y A368849 T(n, n - 1) = 2*A081133(n) for n >= 1.
%Y A368849 Sum_{k=0..n} T(n, k) = A001864(n).
%Y A368849 (Sum_{k=0..n} T(n, k)) / n = A000435(n) for n >= 1.
%Y A368849 (Sum_{k=0..n} T(n, k)) * n / 2 = A262973(n) for n >= 1.
%Y A368849 (Sum_{k=2..n} T(n, k)) / (2*n) = A057500(n) for n >= 1.
%Y A368849 T(n, 1)/(n - 1) + (Sum_{k=2..n} T(n, k)) / (2*n) = A368951(n) for n >= 2.
%Y A368849 Sum_{k=0..n} (-1)^(k-1) * T(n, k) = A368981(n).
%K A368849 nonn,tabl
%O A368849 0,5
%A A368849 _Peter Luschny_, Jan 11 2024