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%I #23 Oct 23 2018 11:09:31
%S 1,1,1,1,0,-1,1,0,2,1,1,0,0,-6,1,1,0,0,6,36,-19,1,0,0,0,-24,-240,151,
%T 1,0,0,0,24,120,1920,-1091,1,0,0,0,0,-120,-360,-17640,7841,1,0,0,0,0,
%U 120,720,0,183120,-56519,1,0,0,0,0,0,-720,-5040,20160,-2116800
%N 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 Seiichi Manyama, <a href="/A293133/b293133.txt">Antidiagonals n = 0..139, flattened</a>
%F A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = (-1)^k * Sum_{i=k..n-1} (-1)^i*(i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.
%e Square array begins:
%e 1, 1, 1, 1, ...
%e 1, 0, 0, 0, ...
%e -1, 2, 0, 0, ...
%e 1, -6, 6, 0, ...
%e 1, 36, -24, 24, ...
%e -19, -240, 120, -120, ...
%o (Ruby)
%o def f(n)
%o return 1 if n < 2
%o (1..n).inject(:*)
%o end
%o def ncr(n, r)
%o return 1 if r == 0
%o (n - r + 1..n).inject(:*) / (1..r).inject(:*)
%o end
%o def A(k, n)
%o ary = [1]
%o (1..n).each{|i| ary << (-1) ** (k % 2) * (k..i - 1).inject(0){|s, j| s + (-1) ** (j % 2) * f(j + 1) * ncr(i - 1, j) * ary[i - 1 - j]}}
%o ary
%o end
%o def A293133(n)
%o a = []
%o (0..n).each{|i| a << A(i, n - i)}
%o ary = []
%o (0..n).each{|i|
%o (0..i).each{|j|
%o ary << a[i - j][j]
%o }
%o }
%o ary
%o end
%o p A293133(20)
%Y Columns k=0..2 give A111884, A293120, A293121.
%Y Rows n=0..1 give A000012, A000007.
%Y Main diagonal gives A000007.
%Y A(n,n-1) gives A000142(n).
%Y Cf. A293053, A293119, A293134,
%K sign,tabl
%O 0,9
%A _Seiichi Manyama_, Sep 30 2017
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