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Triangle read by rows, T(n, k) = (n - k)!*(n - 1)^k, for 0 <= k <= n.
3

%I #7 Dec 25 2021 14:53:09

%S 1,1,0,2,1,1,6,4,4,8,24,18,18,27,81,120,96,96,128,256,1024,720,600,

%T 600,750,1250,3125,15625,5040,4320,4320,5184,7776,15552,46656,279936,

%U 40320,35280,35280,41160,57624,100842,235298,823543,5764801

%N Triangle read by rows, T(n, k) = (n - k)!*(n - 1)^k, for 0 <= k <= n.

%e Triangle starts:

%e [0] 1

%e [1] 1, 0

%e [2] 2, 1, 1

%e [3] 6, 4, 4, 8

%e [4] 24, 18, 18, 27, 81

%e [5] 120, 96, 96, 128, 256, 1024

%e [6] 720, 600, 600, 750, 1250, 3125, 15625

%e [7] 5040, 4320, 4320, 5184, 7776, 15552, 46656, 279936

%e [8] 40320, 35280, 35280, 41160, 57624, 100842, 235298, 823543, 5764801

%p A350269 := (n, k) -> (n - k)!*(n - 1)^k:

%p seq(seq(A350269(n, k), k = 0..n), n = 0..9);

%t T[n_, k_] := If[n - 1 == k == 0, 1, (n - k)! * (n - 1)^k]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Dec 25 2021 *)

%Y Cf. A000142 (first column), A001563 (second column), A000312 (subdiagonal), A065440 (main diagonal), A350268 (row sums).

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, Dec 25 2021