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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).
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%I #19 May 09 2020 02:41:34

%S 1,1,-1,1,-1,1,1,-1,-1,-1,1,-1,-1,5,1,1,-1,-1,-1,1,-1,1,-1,-1,-1,25,

%T -41,1,1,-1,-1,-1,1,19,31,-1,1,-1,-1,-1,1,139,-209,461,1,1,-1,-1,-1,1,

%U 19,151,-2269,-895,-1,1,-1,-1,-1,1,19,871,-1429,2801,-6481,1,1,-1,-1,-1,1,19,151,1091,-19039,68615,22591,-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).

%H Seiichi Manyama, <a href="/A334561/b334561.txt">Antidiagonals n = 0..139, flattened</a>

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

%e -1, 5, -1, -1, -1, -1, -1, ...

%e 1, 1, 25, 1, 1, 1, 1, ...

%e -1, -41, 19, 139, 19, 19, 19, ...

%e 1, 31, -209, 151, 871, 151, 151, ...

%Y Columns k=1..5 give A033999, A000321, A334562, A334564, A334565.

%Y Main diagonal gives A293116.

%Y Cf. A293669, A334568.

%K sign,tabl

%O 0,14

%A _Seiichi Manyama_, May 06 2020