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A293051 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{i=0..k} x^i/i! - exp(x)).

%I A293051 #15 Sep 30 2017 03:10:15
%S A293051 1,1,-1,1,0,0,1,0,-1,1,1,0,0,-1,1,1,0,0,-1,2,-2,1,0,0,0,-1,9,-9,1,0,0,
%T A293051 0,-1,-1,9,-9,1,0,0,0,0,-1,9,-50,50,1,0,0,0,0,-1,-1,34,-267,267,1,0,0,
%U A293051 0,0,0,-1,-1,90,-413,413,1,0,0,0,0,0,-1,-1,34,-71
%N A293051 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(Sum_{i=0..k} x^i/i! - exp(x)).
%H A293051 Seiichi Manyama, <a href="/A293051/b293051.txt">Antidiagonals n = 0..139, flattened</a>
%F A293051 E.g.f. of column k: Product_{i>k} exp(-x^i/i!).
%F A293051 A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = - Sum_{i=k..n-1} binomial(n-1,i)*A(n-1-i,k) for n > k.
%e A293051 Square array begins:
%e A293051     1,  1,  1,  1,  1, ...
%e A293051    -1,  0,  0,  0,  0, ...
%e A293051     0, -1,  0,  0,  0, ...
%e A293051     1, -1, -1,  0,  0, ...
%e A293051     1,  2, -1, -1,  0, ...
%e A293051    -2,  9, -1, -1, -1, ...
%Y A293051 Columns k=0..4 give A000587, A293037, A293038, A293039, A293040.
%Y A293051 Rows n=0..1 give A000012, (-1)*A000007.
%Y A293051 Main diagonal gives A000007.
%Y A293051 Cf. A229223, A293024.
%K A293051 sign,tabl
%O A293051 0,20
%A A293051 _Seiichi Manyama_, Sep 29 2017