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A[0, _] = 0; A[1, _] = 1; A[n_, k_] := A[n, k] = ((n-1)^k + n^k) A[n-1, k] - (n-1)^(2k) A[n-2, k];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, May 11 2019 *)
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Seiichi Manyama, <a href="/A291556/b291556.txt">Table of n, a(n) for Antidiagonals n = 0..182959, flattened</a>
Seiichi Manyama, <a href="/A291556/b291556.txt">Table of n, a(n) for n = 0..1829</a>
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