A354797 Triangle read by rows. T(n, k) = |Stirling1(n, k)| * Stirling2(n + k, n) = A132393(n, k) * A048993(n + k, n).

1, 0, 1, 0, 3, 7, 0, 12, 75, 90, 0, 60, 715, 2100, 1701, 0, 360, 7000, 36750, 69510, 42525, 0, 2520, 72884, 595350, 1940295, 2692305, 1323652, 0, 20160, 814968, 9549120, 47030445, 109794300, 120023904, 49329280, 0, 181440, 9801000, 156008160, 1076453763, 3723239520, 6733767040, 6065579520, 2141764053

Offset: 0

Author

Peter Luschny, Jun 06 2022

Keywords

• nonn
• tabl

Examples

			Table T(n, k) begins:
[0] 1
[1] 0,     1
[2] 0,     3,      7
[3] 0,    12,     75,      90
[4] 0,    60,    715,    2100,     1701
[5] 0,   360,   7000,   36750,    69510,     42525
[6] 0,  2520,  72884,  595350,  1940295,   2692305,   1323652
[7] 0, 20160, 814968, 9549120, 47030445, 109794300, 120023904, 49329280
		

Links

• Mike Earnest, Counting endofunctions by inclusion-exclusion, at Math.StackExchange.

Crossrefs

Cf. A048993, A001710, A007820, A132393, A000312, A354977.

Programs

Maple

T := (n, k) -> abs(Stirling1(n, k))*Stirling2(n + k, n):
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;

Formula

Sum_{k=0..n} (-1)^(n - k)*T(n, k) = n^n. - Werner Schulte, Jun 03 2022 in A000312. [Formerly a conjecture, now proved by Mike Earnest, see link.]

T(n, k) = A132393(n, k) * A354977(n, k) = (1/n!) * Sum_{j=0..n} (-1)^(j + k) * binomial(n, j) * Stirling1(n, k) * j^(n + k).