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%I #34 May 07 2021 05:09:45
%S 1,1,1,1,1,1,1,1,0,1,1,1,-1,-1,1,1,1,-2,-1,1,1,1,1,-3,1,9,2,1,1,1,-4,
%T 5,19,-23,-9,1,1,1,-5,11,25,-128,-25,9,1,1,1,-6,19,21,-343,379,583,50,
%U 1,1,1,-7,29,1,-674,2133,1549,-3087,-267,1
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} (-k)^(n-j)*Stirling2(n,j).
%H Seiichi Manyama, <a href="/A309386/b309386.txt">Antidiagonals n = 0..139, flattened</a>
%F E.g.f. of column k: exp((1 - exp(-k*x))/k) for k > 0.
%F A(0,k) = 1 and A(n,k) = Sum_{j=0..n-1} (-k)^(n-1-j) * binomial(n-1,j) * A(j,k) for n > 0.
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 0, -1, -2, -3, -4, -5, ...
%e 1, -1, -1, 1, 5, 11, 19, ...
%e 1, 1, 9, 19, 25, 21, 1, ...
%e 1, 2, -23, -128, -343, -674, -1103, ...
%e 1, -9, -25, 379, 2133, 6551, 15211, ...
%t T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-j)] * StirlingS2[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Amiram Eldar_, May 07 2021 *)
%Y Columns k=0..6 give A000012, (-1)^n * A000587(n), A009235, A317996, A318179, A318180, A318181.
%Y Rows n=0+1, 2 give A000012, A024000.
%Y Main diagonal gives A318183.
%Y Cf. A241578, A292861.
%K sign,tabl
%O 0,18
%A _Seiichi Manyama_, Jul 27 2019
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