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.
Triangle T(n, k) starts: [0] 1; [1] -1, 1; [2] 2, -2, 1; [3] 0, 12, -3, 1; [4] -56, -120, 28, -4, 1; [5] 0, 1680, -450, 50, -5, 1; [6] 15840, -30240, 10416, -1080, 78, -6, 1; [7] 0, 665280, -317520, 33712, -2100, 112, -7, 1; [8] -17297280, -17297280, 12070080, -1391040, 81648, -3600, 152, -8, 1;
A357340 := proc(n, k) local u; u := n - k; (2*u)!*add(binomial(-n, j) * j! * 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);
# using function A268438 def A357340(n, k): return sum(binomial(-n, i) * A268438(n - k, i) for i in range(n - k + 1)) for n in range(10): print([A357340(n, k) for k in range(n + 1)])