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%I #12 Dec 10 2023 09:23:51
%S 1,-1,1,10,-2,1,-270,24,-3,1,14056,-720,44,-4,1,-1197000,40320,-1500,
%T 70,-5,1,151169040,-3628800,92064,-2700,102,-6,1,-26521775280,
%U 479001600,-8890560,181888,-4410,140,-7,1,6169461217920,-87178291200,1241982720,-18910080,324912,-6720,184,-8,1
%N Triangle read by rows. T(n, k) = Sum_{j=0..n-k} binomial(-n, j) * A268437(n - k, j).
%e Triangle starts:
%e [0] 1;
%e [1] -1, 1;
%e [2] 10, -2, 1;
%e [3] -270, 24, -3, 1;
%e [4] 14056, -720, 44, -4, 1;
%e [5] -1197000, 40320, -1500, 70, -5, 1;
%e [6] 151169040, -3628800, 92064, -2700, 102, -6, 1;
%p A357339 := proc(n, k) local u; u:=(n - k); (2*u)!*add(binomial(-n, j) * j! * add((-1)^(j+m)*binomial(u+j, u+m)*Stirling2(u+m, m), m=0..j) / (u+j)!, j=0..u) end: seq(print(seq(A357339(n, k), k=0..n)), n=0..6);
%o (SageMath) # using function A268437.
%o def A357339(n, k):
%o return sum(binomial(-n, i) * A268437(n - k, i) for i in range(n - k + 1))
%o for n in range(9): print([A357339(n, k) for k in range(n + 1)])
%Y Cf. A357342 (alternating row sums), A268437, A357340.
%K sign,tabl
%O 0,4
%A _Peter Luschny_, Sep 25 2022
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