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A334561 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).

%I A334561 #19 May 09 2020 02:41:34
%S A334561 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 A334561 -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 A334561 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 A334561 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 A334561 Seiichi Manyama, <a href="/A334561/b334561.txt">Antidiagonals n = 0..139, flattened</a>
%F A334561 A(0,k) = 1 and A(n,k) = - (n-1)! * Sum_{j=1..min(k,n)} j*A(n-j,k)/(n-j)!.
%e A334561 Square array begins:
%e A334561    1,   1,    1,   1,   1,   1,   1, ...
%e A334561   -1,  -1,   -1,  -1,  -1,  -1,  -1, ...
%e A334561    1,  -1,   -1,  -1,  -1,  -1,  -1, ...
%e A334561   -1,   5,   -1,  -1,  -1,  -1,  -1, ...
%e A334561    1,   1,   25,   1,   1,   1,   1, ...
%e A334561   -1, -41,   19, 139,  19,  19,  19, ...
%e A334561    1,  31, -209, 151, 871, 151, 151, ...
%Y A334561 Columns k=1..5 give A033999, A000321, A334562, A334564, A334565.
%Y A334561 Main diagonal gives A293116.
%Y A334561 Cf. A293669, A334568.
%K A334561 sign,tabl
%O A334561 0,14
%A A334561 _Seiichi Manyama_, May 06 2020