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A291709 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).

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