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 A344913 #14 Aug 14 2021 15:49:31
%S A344913 1,0,1,0,-2,2,0,4,-12,6,0,-8,56,-72,24,0,16,-240,600,-480,120,0,-32,
%T A344913 992,-4320,6240,-3600,720,0,64,-4032,28896,-67200,67200,-30240,5040,0,
%U A344913 -128,16256,-185472,653184,-1008000,766080,-282240,40320
%N A344913 Table read by rows, T(n, k) (for 0 <= k <= n) = (-2)^(n - k)*k!*Stirling2(n, k).
%F A344913 T(n, k) = 2^(n - k)*Sum_{j=0..n} (-1)^(n - j)*binomial(k, j)*j^n.
%F A344913 Let row(n, x) be the n-th row polynomial, then B(n) = row(n-1, 1)*n / (4^n - 2^n) is the n-th Bernoulli number (with B(1) = 1/2) for n >= 1.
%e A344913 Table starts:
%e A344913 [0] 1;
%e A344913 [1] 0, 1;
%e A344913 [2] 0, -2, 2;
%e A344913 [3] 0, 4, -12, 6;
%e A344913 [4] 0, -8, 56, -72, 24;
%e A344913 [5] 0, 16, -240, 600, -480, 120;
%e A344913 [6] 0, -32, 992, -4320, 6240, -3600, 720;
%e A344913 [7] 0, 64, -4032, 28896, -67200, 67200, -30240, 5040;
%e A344913 [8] 0, -128, 16256, -185472, 653184, -1008000, 766080, -282240, 40320.
%p A344913 T := (n, k) -> (-2)^(n - k)*k!*Stirling2(n, k):
%p A344913 seq(seq(T(n, k), k = 0..n), n = 0..9);
%o A344913 (PARI) T(n, k) = (-2)^(n - k)*k!*stirling(n, k, 2); \\ _Michel Marcus_, Aug 14 2021
%Y A344913 Cf. A155585 (row sums), A122704 (alternating row sums, signed), A278075 (signed Fubini polynomials), A000142 (main diagonal), A048993 (Stirling2).
%K A344913 sign,tabl
%O A344913 0,5
%A A344913 _Peter Luschny_, Aug 14 2021