A292973 - OEIS

%I #27 Jul 10 2022 09:40:59

%S 1,1,-1,1,1,2,1,0,-1,-5,1,0,2,-2,15,1,0,0,-6,9,-52,1,0,0,6,24,-4,203,

%T 1,0,0,0,-24,-140,-95,-877,1,0,0,0,24,60,870,414,4140,1,0,0,0,0,-120,

%U 240,-5922,49,-21147,1,0,0,0,0,120,360,-4830,45416,-10088,115975

%N Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (-1)^(k+1) * k! * Sum_{i=0..n-1} (-1)^i * binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.

%H Seiichi Manyama, <a href="/A292973/b292973.txt">Antidiagonals n = 0..139, flattened</a>

%F T(n,k) = n! * Sum_{j=0..floor(n/k)} (-j)^(n-k*j)/(j! * (n-k*j)!) for k > 0. - _Seiichi Manyama_, Jul 10 2022

%e Square array begins:

%e    1,  1,  1,   1,  1, ...

%e   -1,  1,  0,   0,  0, ...

%e    2, -1,  2,   0,  0, ...

%e   -5, -2, -6,   6,  0, ...

%e   15,  9, 24, -24, 24, ...

%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 + 1) * f(k) * (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 A292973(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 A292973(20)

%Y Columns k=0-5 give: A292935, A003725, A292907, A292908, A292969, A292970.

%Y Rows n=0 gives A000012.

%Y Main diagonal gives A000142.

%Y Cf. A292948, A292978.

%K sign,tabl

%O 0,6

%A _Seiichi Manyama_, Sep 27 2017