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 A351776 #15 Feb 19 2022 13:58:15
%S A351776 1,1,0,1,-1,0,1,-2,0,0,1,-3,4,3,0,1,-4,12,-6,-4,0,1,-5,24,-63,-8,-25,
%T A351776 0,1,-6,40,-204,420,150,114,0,1,-7,60,-465,2288,-3435,-972,287,0,1,-8,
%U A351776 84,-882,7180,-32020,33462,3682,-4152,0,1,-9,112,-1491,17256,-138525,537576,-379155,6256,1647,0
%N A351776 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * (n-j)^j/j!.
%F A351776 E.g.f. of column k: 1/(1 + k*x*exp(x)).
%F A351776 T(0,k) = 1 and T(n,k) = -k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.
%e A351776 Square array begins:
%e A351776 1, 1, 1, 1, 1, 1, ...
%e A351776 0, -1, -2, -3, -4, -5, ...
%e A351776 0, 0, 4, 12, 24, 40, ...
%e A351776 0, 3, -6, -63, -204, -465, ...
%e A351776 0, -4, -8, 420, 2288, 7180, ...
%e A351776 0, -25, 150, -3435, -32020, -138525, ...
%o A351776 (PARI) T(n, k) = n!*sum(j=0, n, (-k)^(n-j)*(n-j)^j/j!);
%o A351776 (PARI) T(n, k) = if(n==0, 1, -k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));
%Y A351776 Columns k=0..3 give A000007, A302397, A351777, A351778.
%Y A351776 Main diagonal gives A351779.
%Y A351776 Cf. A292861, A351761.
%K A351776 sign,tabl
%O A351776 0,8
%A A351776 _Seiichi Manyama_, Feb 19 2022