%I #22 Oct 23 2018 19:18:10
%S 1,1,-1,1,1,2,1,0,-1,-5,1,0,1,-2,15,1,0,0,-3,9,-52,1,0,0,1,9,-4,203,1,
%T 0,0,0,-4,-40,-95,-877,1,0,0,0,1,10,210,414,4140,1,0,0,0,0,-5,-10,
%U -1176,49,-21147,1,0,0,0,0,1,15,-105,7273,-10088,115975,1,0,0
%N Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = (-1)^(k+1) * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0.
%H Seiichi Manyama, <a href="/A292948/b292948.txt">Antidiagonals n = 0..139, flattened</a>
%e Square array begins:
%e 1, 1, 1, 1, 1, ...
%e -1, 1, 0, 0, 0, ...
%e 2, -1, 1, 0, 0, ...
%e -5, -2, -3, 1, 0, ...
%e 15, 9, 9, -4, 1, ...
%o (Ruby)
%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 + 1) * (0..i - 1).inject(0){|s, j| s + (-1) ** (j % 2) * ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
%o ary
%o end
%o def A292948(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 A292948(20)
%Y Columns k=0-5 give: A292935, A003725, A292909, A292910, A292912, A292950.
%Y Rows n=0 gives A000012.
%Y Main diagonal gives A000012.
%Y Cf. A145460.
%K sign,tabl,look
%O 0,6
%A _Seiichi Manyama_, Sep 27 2017