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A294046 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(1/(1-x)^k - 1).

%I A294046 #21 Nov 03 2017 11:53:25
%S A294046 1,1,0,1,1,0,1,2,3,0,1,3,10,13,0,1,4,21,68,73,0,1,5,36,195,580,501,0,
%T A294046 1,6,55,424,2241,5912,4051,0,1,7,78,785,6136,30483,69784,37633,0,1,8,
%U A294046 105,1308,13705,104544,476469,933200,394353,0,1,9,136,2023,26748
%N A294046 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(1/(1-x)^k - 1).
%H A294046 Seiichi Manyama, <a href="/A294046/b294046.txt">Antidiagonals n = 0..139, flattened</a>
%F A294046 A(0,k) = 1 and A(n,k) = k * (n-1)! * Sum_{j=1..n} binomial(j+k-1,k)*A(n-j,k)/(n-j)! for n > 0.
%e A294046 Square array A(n,k) begins:
%e A294046    1,   1,    1,     1,      1, ...
%e A294046    0,   1,    2,     3,      4, ...
%e A294046    0,   3,   10,    21,     36, ...
%e A294046    0,  13,   68,   195,    424, ...
%e A294046    0,  73,  580,  2241,   6136, ...
%e A294046    0, 501, 5912, 30483, 104544, ...
%t A294046 A[0, _] = 1; A[n_, k_] := k*(n-1)!*Sum[Binomial[j+k-1, k]*A[n-j, k]/(n-j)!, {j, 1, n}];
%t A294046 Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Nov 03 2017 *)
%Y A294046 Columns k=0..5 give A000007, A000262, A136658, A202826, A294050, A294051.
%Y A294046 Rows n=0..2 give A000012, A001477, A014105.
%Y A294046 Main diagonal gives A294047.
%Y A294046 Cf. A291709.
%K A294046 nonn,tabl
%O A294046 0,8
%A A294046 _Seiichi Manyama_, Oct 22 2017