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).

1, 0, 1, 0, 3, 2, 0, 13, 18, 6, 0, 73, 158, 108, 24, 0, 501, 1510, 1590, 720, 120, 0, 4051, 15962, 23040, 15960, 5400, 720, 0, 37633, 186270, 345786, 325920, 168000, 45360, 5040, 0, 394353, 2385182, 5469492, 6579384, 4594800, 1884960, 423360, 40320

Offset: 0

Author

Peter Luschny, Sep 01 2022

Keywords

• nonn
• tabl

Comments

The same construction with Stirling1 in place of Stirling2 gives A225479, the ordered Stirling cycle numbers.

Examples

			Triangle T(n, k) begins:
[0] 1;
[1] 0,      1;
[2] 0,      3,       2;
[3] 0,     13,      18,       6;
[4] 0,     73,     158,     108,      24;
[5] 0,    501,    1510,    1590,     720,     120;
[6] 0,   4051,   15962,   23040,   15960,    5400,     720;
[7] 0,  37633,  186270,  345786,  325920,  168000,   45360,   5040;
[8] 0, 394353, 2385182, 5469492, 6579384, 4594800, 1884960, 423360, 40320;
		

Crossrefs

Cf. A271703, A048993, A225479, A000262 (column 1), A052838 (column 2), A084358 (row sums).

Programs

Maple

L := (n, k) -> `if`(n = k, 1, binomial(n-1, k-1) * n! / k!):
T := (n, k) -> k! * add(L(n, j) * Stirling2(j, k), j = k..n):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Mathematica

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 *)