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A356654 Triangle read by rows. T(n, k) = k! * Sum_{j=k..n} Lah(n, j) * Stirling2(j, k), where Lah(n, k) = A271703(n, k).

%I A356654 #8 Sep 01 2022 17:29:13
%S A356654 1,0,1,0,3,2,0,13,18,6,0,73,158,108,24,0,501,1510,1590,720,120,0,4051,
%T A356654 15962,23040,15960,5400,720,0,37633,186270,345786,325920,168000,45360,
%U A356654 5040,0,394353,2385182,5469492,6579384,4594800,1884960,423360,40320
%N A356654 Triangle read by rows. T(n, k) = k! * Sum_{j=k..n} Lah(n, j) * Stirling2(j, k), where Lah(n, k) = A271703(n, k).
%C A356654 The same construction with Stirling1 in place of Stirling2 gives A225479, the ordered Stirling cycle numbers.
%e A356654 Triangle T(n, k) begins:
%e A356654 [0] 1;
%e A356654 [1] 0,      1;
%e A356654 [2] 0,      3,       2;
%e A356654 [3] 0,     13,      18,       6;
%e A356654 [4] 0,     73,     158,     108,      24;
%e A356654 [5] 0,    501,    1510,    1590,     720,     120;
%e A356654 [6] 0,   4051,   15962,   23040,   15960,    5400,     720;
%e A356654 [7] 0,  37633,  186270,  345786,  325920,  168000,   45360,   5040;
%e A356654 [8] 0, 394353, 2385182, 5469492, 6579384, 4594800, 1884960, 423360, 40320;
%p A356654 L := (n, k) -> `if`(n = k, 1, binomial(n-1, k-1) * n! / k!):
%p A356654 T := (n, k) -> k! * add(L(n, j) * Stirling2(j, k), j = k..n):
%p A356654 seq(seq(T(n, k), k = 0..n), n = 0..9);
%t A356654 T[n_, k_] := k! * Sum[Binomial[n, j] * FactorialPower[n - 1, n - j] * StirlingS2[j, k], {j, k, n}]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Amiram Eldar_, Sep 01 2022 *)
%Y A356654 Cf. A271703, A048993, A225479, A000262 (column 1), A052838 (column 2), A084358 (row sums).
%K A356654 nonn,tabl
%O A356654 0,5
%A A356654 _Peter Luschny_, Sep 01 2022