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Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j/j).
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%I #17 May 09 2020 02:41:37

%S 1,1,-1,1,-1,1,1,-1,0,-1,1,-1,0,2,1,1,-1,0,0,-2,-1,1,-1,0,0,6,-6,1,1,

%T -1,0,0,0,-6,16,-1,1,-1,0,0,0,24,-24,20,1,1,-1,0,0,0,0,-24,-120,-132,

%U -1,1,-1,0,0,0,0,120,-120,540,-28,1,1,-1,0,0,0,0,0,-120,-720,1764,1216,-1

%N Square array A(n,k), n >= 0, k >= 1, read by antidiagonals downwards, where column k is the expansion of e.g.f. exp(-Sum_{j=1..k} x^j/j).

%H Seiichi Manyama, <a href="/A334568/b334568.txt">Antidiagonals n = 0..139, flattened</a>

%F A(0,k) = 1 and A(n,k) = - (n-1)! * Sum_{j=1..min(k,n)} A(n-j,k)/(n-j)!.

%e Square array begins:

%e 1, 1, 1, 1, 1, 1, 1, ...

%e -1, -1, -1, -1, -1, -1, -1, ...

%e 1, 0, 0, 0, 0, 0, 0, ...

%e -1, 2, 0, 0, 0, 0, 0, ...

%e 1, -2, 6, 0, 0, 0, 0, ...

%e -1, -6, -6, 24, 0, 0, 0, ...

%e 1, 16, -24, -24, 120, 0, 0, ...

%e -1, 20, -120, -120, -120, 720, 0, ...

%e 1, -132, 540, -720, -720, -720, 5040, ...

%Y Columns k=1..5 give A033999, A001464, A334569, A334570, A334571.

%Y Cf. A330858, A334561.

%K sign,tabl

%O 0,14

%A _Seiichi Manyama_, May 06 2020