This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357340 #13 Dec 10 2023 09:23:38
%S A357340 1,-1,1,2,-2,1,0,12,-3,1,-56,-120,28,-4,1,0,1680,-450,50,-5,1,15840,
%T A357340 -30240,10416,-1080,78,-6,1,0,665280,-317520,33712,-2100,112,-7,1,
%U A357340 -17297280,-17297280,12070080,-1391040,81648,-3600,152,-8,1
%N A357340 Triangle read by rows. T(n, k) = Sum_{j=0..n-k} binomial(-n, j) * A268438(n - k, j).
%e A357340 Triangle T(n, k) starts:
%e A357340 [0] 1;
%e A357340 [1] -1, 1;
%e A357340 [2] 2, -2, 1;
%e A357340 [3] 0, 12, -3, 1;
%e A357340 [4] -56, -120, 28, -4, 1;
%e A357340 [5] 0, 1680, -450, 50, -5, 1;
%e A357340 [6] 15840, -30240, 10416, -1080, 78, -6, 1;
%e A357340 [7] 0, 665280, -317520, 33712, -2100, 112, -7, 1;
%e A357340 [8] -17297280, -17297280, 12070080, -1391040, 81648, -3600, 152, -8, 1;
%p A357340 A357340 := proc(n, k) local u; u := n - k; (2*u)!*add(binomial(-n, j) * j! *
%p A357340 add((-1)^(j+m)*binomial(u+j, u+m)*abs(Stirling1(u+m, m)), m=0..j)/(u +j)!, j=0..u) end: seq(print(seq(A357340(n, k), k=0..n)), n=0..8);
%o A357340 (SageMath) # using function A268438
%o A357340 def A357340(n, k):
%o A357340 return sum(binomial(-n, i) * A268438(n - k, i) for i in range(n - k + 1))
%o A357340 for n in range(10): print([A357340(n, k) for k in range(n + 1)])
%Y A357340 Cf. A357341 (alternating row sums), A264437, A268438, A357339.
%K A357340 sign,tabl
%O A357340 0,4
%A A357340 _Peter Luschny_, Sep 25 2022