A379821 Array read by ascending antidiagonals: A(n, k) = (-1)^(n + k) * Sum_{j=0..k} (j!)^2 * Stirling1(n, j) * Stirling1(k, j).
1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 5, 2, 0, 0, 6, 14, 14, 6, 0, 0, 24, 50, 76, 50, 24, 0, 0, 120, 224, 360, 360, 224, 120, 0, 0, 720, 1216, 1908, 2392, 1908, 1216, 720, 0, 0, 5040, 7776, 11628, 15664, 15664, 11628, 7776, 5040, 0
Offset: 0
Examples
Array begins: [0] 1, 0, 0, 0, 0, 0, 0, 0, ... [1] 0, 1, 1, 2, 6, 24, 120, 720, ... [2] 0, 1, 5, 14, 50, 224, 1216, 7776, ... [3] 0, 2, 14, 76, 360, 1908, 11628, 81072, ... [4] 0, 6, 50, 360, 2392, 15664, 110336, 856080, ... [5] 0, 24, 224, 1908, 15664, 126676, 1046780, 9169920, ... [6] 0, 120, 1216, 11628, 110336, 1046780, 10057204, 99846144, ... . Triangle T(n, k) = A(n - k, k) starts: [0] 1; [1] 0, 0; [2] 0, 1, 0; [3] 0, 1, 1, 0; [4] 0, 2, 5, 2, 0; [5] 0, 6, 14, 14, 6, 0; [6] 0, 24, 50, 76, 50, 24, 0; [7] 0, 120, 224, 360, 360, 224, 120, 0; [8] 0, 720, 1216, 1908, 2392, 1908, 1216, 720, 0;
Programs
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Maple
A := (n, k) -> local j; (-1)^(n + k)*add((j!)^2*Stirling1(n, j)*Stirling1(k, j), j = 0..k): seq(lprint(seq(A(n, k), k = 0..7)), n = 0..8);
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PARI
a(n, k) = sum(j=0, min(n, k), j!^2*abs(stirling(n, j, 1)*stirling(k, j, 1))); \\ Seiichi Manyama, Apr 05 2025
Formula
E.g.f.: 1 / (1 - log(1-x) * log(1-y)). - Seiichi Manyama, Apr 05 2025