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 A351761 #21 Feb 19 2022 13:58:42
%S A351761 1,1,0,1,1,0,1,2,4,0,1,3,12,21,0,1,4,24,102,148,0,1,5,40,279,1160,
%T A351761 1305,0,1,6,60,588,4332,16490,13806,0,1,7,84,1065,11536,84075,281292,
%U A351761 170401,0,1,8,112,1746,25220,282900,1958058,5598110,2403640,0
%N A351761 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 A351761 E.g.f. of column k: 1/(1 - k*x*exp(x)).
%F A351761 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 A351761 Square array begins:
%e A351761 1, 1, 1, 1, 1, 1, ...
%e A351761 0, 1, 2, 3, 4, 5, ...
%e A351761 0, 4, 12, 24, 40, 60, ...
%e A351761 0, 21, 102, 279, 588, 1065, ...
%e A351761 0, 148, 1160, 4332, 11536, 25220, ...
%e A351761 0, 1305, 16490, 84075, 282900, 746525, ...
%o A351761 (PARI) T(n, k) = n!*sum(j=0, n, k^(n-j)*(n-j)^j/j!);
%o A351761 (PARI) T(n, k) = if(n==0, 1, k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));
%Y A351761 Columns k=0..3 give A000007, A006153, A351762, A351763.
%Y A351761 Main diagonal gives A351765.
%Y A351761 Cf. A292860, A351776.
%K A351761 nonn,tabl
%O A351761 0,8
%A A351761 _Seiichi Manyama_, Feb 18 2022