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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).

%I A369016 #22 Jan 28 2024 18:07:41
%S A369016 0,0,0,0,1,0,0,6,2,0,0,48,18,12,0,0,500,192,144,108,0,0,6480,2500,
%T A369016 1920,1620,1280,0,0,100842,38880,30000,25920,23040,18750,0,0,1835008,
%U A369016 705894,544320,472500,430080,393750,326592,0
%N 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).
%F A369016 T = B066320 - A369017 (where B066320 = A066320 after adding a 0-column to the left and then setting offset to (0, 0)).
%e A369016 Triangle starts:
%e A369016   [0] [0]
%e A369016   [1] [0,       0]
%e A369016   [2] [0,       1,      0]
%e A369016   [3] [0,       6,      2,      0]
%e A369016   [4] [0,      48,     18,     12,      0]
%e A369016   [5] [0,     500,    192,    144,    108,      0]
%e A369016   [6] [0,    6480,   2500,   1920,   1620,   1280,      0]
%e A369016   [7] [0,  100842,  38880,  30000,  25920,  23040,  18750,      0]
%e A369016   [8] [0, 1835008, 705894, 544320, 472500, 430080, 393750, 326592, 0]
%p A369016 T := (n, k) -> binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1):
%p A369016 seq(seq(T(n, k), k = 0..n), n=0..9);
%t A369016 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 *)
%o A369016 (SageMath)
%o A369016 def T(n, k): return binomial(n-1, k-1)*(k-1)^(k-1)*(n-k)*(n-k+1)^(n-k-1)
%o A369016 for n in range(0, 9): print([T(n, k) for k in range(n + 1)])
%Y A369016 A368849, A368982 and this sequence are alternative sum representation for A001864 with different normalizations.
%Y A369016 T(n, k) = A368849(n, k) / n for n >= 1.
%Y A369016 T(n, 1) = A053506(n) for n >= 1.
%Y A369016 T(n, n - 1) = A055897(n - 1) for n >= 2.
%Y A369016 Sum_{k=0..n} T(n, k) = A000435(n) for n >= 1.
%Y A369016 Sum_{k=0..n} (-1)^(k+1)*T(n, k) = A368981(n) / n for n >= 1.
%Y A369016 Cf. A066320, A369017.
%K A369016 nonn,tabl
%O A369016 0,8
%A A369016 _Peter Luschny_, Jan 12 2024