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A355007 Triangle read by rows. T(n, k) = n^k * |Stirling1(n, k)|.

%I A355007 #15 Mar 31 2023 05:21:13
%S A355007 1,0,1,0,2,4,0,6,27,27,0,24,176,384,256,0,120,1250,4375,6250,3125,0,
%T A355007 720,9864,48600,110160,116640,46656,0,5040,86436,557032,1764735,
%U A355007 2941225,2470629,823543,0,40320,836352,6723584,27725824,64225280,84410368,58720256,16777216
%N A355007 Triangle read by rows. T(n, k) = n^k * |Stirling1(n, k)|.
%F A355007 Sum_{k=0..n} (-1)^k * T(n,k) = A133942(n). - _Alois P. Heinz_, Mar 30 2023
%F A355007 Conjecture: T(n,k) = A056856(n,k)*n. - _R. J. Mathar_, Mar 31 2023
%e A355007 Table T(n, k) begins:
%e A355007 [0] 1;
%e A355007 [1] 0,    1;
%e A355007 [2] 0,    2,     4;
%e A355007 [3] 0,    6,    27,     27;
%e A355007 [4] 0,   24,   176,    384,     256;
%e A355007 [5] 0,  120,  1250,   4375,    6250,    3125;
%e A355007 [6] 0,  720,  9864,  48600,  110160,  116640,   46656;
%e A355007 [7] 0, 5040, 86436, 557032, 1764735, 2941225, 2470629, 823543;
%p A355007 seq(seq(n^k*abs(Stirling1(n, k)), k = 0..n), n = 0..9);
%t A355007 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 A355007 A000142 (column 1), A000407 (row sums), A000312 (main diagonal), A355006.
%Y A355007 Cf. A133942.
%K A355007 nonn,tabl
%O A355007 0,5
%A A355007 _Peter Luschny_, Jun 17 2022