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%I #33 Nov 03 2017 11:53:41
%S 1,1,1,1,1,1,1,1,2,1,1,1,3,6,1,1,1,4,13,24,1,1,1,5,22,73,120,1,1,1,6,
%T 33,154,501,720,1,1,1,7,46,273,1306,4051,5040,1,1,1,8,61,436,2721,
%U 12976,37633,40320,1,1,1,9,78,649,4956,31701,147484,394353,362880,1
%N Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} (-1)^(j-1)*binomial(-k,j-1)*x^j/j).
%H Seiichi Manyama, <a href="/A291709/b291709.txt">Antidiagonals n = 0..139, flattened</a>
%F Let B(j,k) = (-1)^(j-1)*binomial(-k,j-1) for j>0 and k>=0.
%F A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} B(j,k)*A(n-j,k)/(n-j)! for n > 0.
%e Square array B(j,k) begins:
%e 1, 1, 1, 1, 1, ...
%e 0, 1, 2, 3, 4, ...
%e 0, 1, 3, 6, 10, ...
%e 0, 1, 4, 10, 20, ...
%e 0, 1, 5, 15, 35, ...
%e 0, 1, 6, 21, 56, ...
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, ...
%e 1, 6, 13, 22, 33, ...
%e 1, 24, 73, 154, 273, ...
%e 1, 120, 501, 1306, 2721, ...
%t B[j_, k_] := (-1)^(j-1)*Binomial[-k, j-1];
%t A[0, _] = 1; A[n_, k_] := (n-1)!*Sum[B[j, k]*A[n-j, k]/(n-j)!, {j, 1, n}];
%t Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Nov 03 2017 *)
%Y Columns k=0..10 give A000012, A000142, A000262, A049376, A049377, A049378, A049402, A132164, A293986, A293987, A293988.
%Y Rows n=0-1 give A000012.
%Y Main diagonal gives A293989.
%Y Cf. A293012, A293991.
%K nonn,tabl
%O 0,9
%A _Seiichi Manyama_, Oct 21 2017
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