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A369019 Triangle read by rows: T(n, k) = binomial(n, k - 1)*(k - 1)^(k - 1)*k*(n - k + 1)^(n - k).

%I A369019 #15 Jan 27 2024 18:48:35
%S A369019 0,0,1,0,2,4,0,9,12,36,0,64,72,144,432,0,625,640,1080,2160,6400,0,
%T A369019 7776,7500,11520,19440,38400,112500,0,117649,108864,157500,241920,
%U A369019 403200,787500,2286144,0,2097152,1882384,2612736,3780000,5734400,9450000,18289152,52706752
%N A369019 Triangle read by rows: T(n, k) = binomial(n, k - 1)*(k - 1)^(k - 1)*k*(n - k + 1)^(n - k).
%F A369019 T = A369018 - A368849.
%e A369019 Triangle starts:
%e A369019 [0] [0]
%e A369019 [1] [0,      1]
%e A369019 [2] [0,      2,      4]
%e A369019 [3] [0,      9,     12,     36]
%e A369019 [4] [0,     64,     72,    144,    432]
%e A369019 [5] [0,    625,    640,   1080,   2160,   6400]
%e A369019 [6] [0,   7776,   7500,  11520,  19440,  38400, 112500]
%e A369019 [7] [0, 117649, 108864, 157500, 241920, 403200, 787500, 2286144]
%p A369019 T := (n, k) -> binomial(n, k - 1)*(k - 1)^(k - 1)*k*(n - k + 1)^(n - k):
%p A369019 seq(seq(T(n, k), k = 0..n), n=0..9);
%t A369019 A369019[n_, k_] := Binomial[n, k-1] If[k == 1, 1, (k-1)^(k-1)] k (n-k+1)^(n-k);
%t A369019 Table[A369019[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Jan 27 2024 *)
%o A369019 (SageMath)
%o A369019 def A369019(n, k):
%o A369019     return binomial(n, k - 1)*(k - 1)^(k - 1)*k*(n - k + 1)^(n - k)
%Y A369019 Cf. A369018, A368849.
%K A369019 nonn,tabl
%O A369019 0,5
%A A369019 _Peter Luschny_, Jan 13 2024