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

%I A354797 #18 Jun 16 2022 14:05:35
%S A354797 1,0,1,0,3,7,0,12,75,90,0,60,715,2100,1701,0,360,7000,36750,69510,
%T A354797 42525,0,2520,72884,595350,1940295,2692305,1323652,0,20160,814968,
%U A354797 9549120,47030445,109794300,120023904,49329280,0,181440,9801000,156008160,1076453763,3723239520,6733767040,6065579520,2141764053
%N A354797 Triangle read by rows. T(n, k) = |Stirling1(n, k)| * Stirling2(n + k, n) = A132393(n, k) * A048993(n + k, n).
%H A354797 Mike Earnest, <a href="https://math.stackexchange.com/a/4473971">Counting endofunctions by inclusion-exclusion</a>, at Math.StackExchange.
%F A354797 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 A354797 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 A354797 Table T(n, k) begins:
%e A354797 [0] 1
%e A354797 [1] 0,     1
%e A354797 [2] 0,     3,      7
%e A354797 [3] 0,    12,     75,      90
%e A354797 [4] 0,    60,    715,    2100,     1701
%e A354797 [5] 0,   360,   7000,   36750,    69510,     42525
%e A354797 [6] 0,  2520,  72884,  595350,  1940295,   2692305,   1323652
%e A354797 [7] 0, 20160, 814968, 9549120, 47030445, 109794300, 120023904, 49329280
%p A354797 T := (n, k) -> abs(Stirling1(n, k))*Stirling2(n + k, n):
%p A354797 for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
%Y A354797 Cf. A048993, A001710, A007820, A132393, A000312, A354977.
%K A354797 nonn,tabl
%O A354797 0,5
%A A354797 _Peter Luschny_, Jun 06 2022