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 A379820 #7 Jan 15 2025 05:52:51
%S A379820 1,0,0,0,1,0,0,1,1,0,0,2,3,2,0,0,6,8,8,6,0,0,24,28,28,28,24,0,0,120,
%T A379820 124,114,114,124,120,0,0,720,668,558,518,558,668,720,0,0,5040,4248,
%U A379820 3234,2744,2744,3234,4248,5040,0,0,40320,31176,21768,16888,15446,16888,21768,31176,40320,0
%N A379820 Array read by ascending antidiagonals: A(n, k) = (-1)^(n + k) * Sum_{j=0..k} j! * Stirling1(n, j) * Stirling1(k, j).
%e A379820 Array begins:
%e A379820 [0] 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A379820 [1] 0, 1, 1, 2, 6, 24, 120, 720, 5040, ...
%e A379820 [2] 0, 1, 3, 8, 28, 124, 668, 4248, 31176, ...
%e A379820 [3] 0, 2, 8, 28, 114, 558, 3234, 21768, 167280, ...
%e A379820 [4] 0, 6, 28, 114, 518, 2744, 16888, 119232, 952944, ...
%e A379820 [5] 0, 24, 124, 558, 2744, 15446, 99730, 732120, 6045240, ...
%e A379820 [6] 0, 120, 668, 3234, 16888, 99730, 669422, 5074992, 43062864, ...
%e A379820 .
%e A379820 Triangle T(n, k) = A(n - k, k) starts:
%e A379820 [0] 1;
%e A379820 [1] 0, 0;
%e A379820 [2] 0, 1, 0;
%e A379820 [3] 0, 1, 1, 0;
%e A379820 [4] 0, 2, 3, 2, 0;
%e A379820 [5] 0, 6, 8, 8, 6, 0;
%e A379820 [6] 0, 24, 28, 28, 28, 24, 0;
%e A379820 [7] 0, 120, 124, 114, 114, 124, 120, 0;
%e A379820 [8] 0, 720, 668, 558, 518, 558, 668, 720, 0;
%p A379820 A := (n, k) -> local j; (-1)^(n + k)*add(j!*Stirling1(n, j)*Stirling1(k, j), j = 0..k):
%p A379820 seq(lprint(seq(A(n, k), k = 0..8)), n = 0..8);
%Y A379820 The corresponding array with Stirling2 numbers is A108470.
%K A379820 nonn,tabl
%O A379820 0,12
%A A379820 _Peter Luschny_, Jan 03 2025