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%I #15 Mar 31 2023 05:21:13
%S 1,0,1,0,2,4,0,6,27,27,0,24,176,384,256,0,120,1250,4375,6250,3125,0,
%T 720,9864,48600,110160,116640,46656,0,5040,86436,557032,1764735,
%U 2941225,2470629,823543,0,40320,836352,6723584,27725824,64225280,84410368,58720256,16777216
%N Triangle read by rows. T(n, k) = n^k * |Stirling1(n, k)|.
%F Sum_{k=0..n} (-1)^k * T(n,k) = A133942(n). - _Alois P. Heinz_, Mar 30 2023
%F Conjecture: T(n,k) = A056856(n,k)*n. - _R. J. Mathar_, Mar 31 2023
%e Table T(n, k) begins:
%e [0] 1;
%e [1] 0, 1;
%e [2] 0, 2, 4;
%e [3] 0, 6, 27, 27;
%e [4] 0, 24, 176, 384, 256;
%e [5] 0, 120, 1250, 4375, 6250, 3125;
%e [6] 0, 720, 9864, 48600, 110160, 116640, 46656;
%e [7] 0, 5040, 86436, 557032, 1764735, 2941225, 2470629, 823543;
%p seq(seq(n^k*abs(Stirling1(n, k)), k = 0..n), n = 0..9);
%t T[n_, k_] := If[n == k == 0, 1, n^k * Abs[StirlingS1[n, k]]]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Jun 17 2022 *)
%Y A000142 (column 1), A000407 (row sums), A000312 (main diagonal), A355006.
%Y Cf. A133942.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Jun 17 2022
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