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A357339 Triangle read by rows. T(n, k) = Sum_{j=0..n-k} binomial(-n, j) * A268437(n - k, j).

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