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A309386 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).

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