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A293134 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-x^(k+1)/(1+x)).

%I A293134 #20 Oct 01 2017 00:27:31
%S A293134 1,1,-1,1,0,3,1,0,-2,-13,1,0,0,6,73,1,0,0,-6,-12,-501,1,0,0,0,24,0,
%T A293134 4051,1,0,0,0,-24,-120,240,-37633,1,0,0,0,0,120,1080,-2520,394353,1,0,
%U A293134 0,0,0,-120,-720,-10080,21840,-4596553,1,0,0,0,0,0,720,5040,100800
%N A293134 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-x^(k+1)/(1+x)).
%H A293134 Seiichi Manyama, <a href="/A293134/b293134.txt">Antidiagonals n = 0..139, flattened</a>
%F A293134 A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = (-1)^(k+1) * Sum_{i=k..n-1} (-1)^i*(i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.
%e A293134 Square array begins:
%e A293134       1,   1,    1,   1, ...
%e A293134      -1,   0,    0,   0, ...
%e A293134       3,  -2,    0,   0, ...
%e A293134     -13,   6,   -6,   0, ...
%e A293134      73, -12,   24, -24, ...
%e A293134    -501,   0, -120, 120, ...
%Y A293134 Columns k=0..2 give A293125, A293122, A293123.
%Y A293134 Rows n=0..1 give A000012, (-1)*A000007.
%Y A293134 Main diagonal gives A000007.
%Y A293134 A(n,n-1) gives (-1)*A000142(n).
%Y A293134 Cf. A293053, A293119, A293133.
%K A293134 sign,tabl
%O A293134 0,6
%A A293134 _Seiichi Manyama_, Sep 30 2017