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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.
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%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