%I #18 Jun 16 2022 14:05:35
%S 1,0,1,0,3,7,0,12,75,90,0,60,715,2100,1701,0,360,7000,36750,69510,
%T 42525,0,2520,72884,595350,1940295,2692305,1323652,0,20160,814968,
%U 9549120,47030445,109794300,120023904,49329280,0,181440,9801000,156008160,1076453763,3723239520,6733767040,6065579520,2141764053
%N Triangle read by rows. T(n, k) = |Stirling1(n, k)| * Stirling2(n + k, n) = A132393(n, k) * A048993(n + k, n).
%H Mike Earnest, <a href="https://math.stackexchange.com/a/4473971">Counting endofunctions by inclusion-exclusion</a>, at Math.StackExchange.
%F 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.]
%F 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).
%e Table T(n, k) begins:
%e [0] 1
%e [1] 0, 1
%e [2] 0, 3, 7
%e [3] 0, 12, 75, 90
%e [4] 0, 60, 715, 2100, 1701
%e [5] 0, 360, 7000, 36750, 69510, 42525
%e [6] 0, 2520, 72884, 595350, 1940295, 2692305, 1323652
%e [7] 0, 20160, 814968, 9549120, 47030445, 109794300, 120023904, 49329280
%p T := (n, k) -> abs(Stirling1(n, k))*Stirling2(n + k, n):
%p for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
%Y Cf. A048993, A001710, A007820, A132393, A000312, A354977.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Jun 06 2022