A292973 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.
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, 1, 0, 0, 0, -24, -140, -95, -877, 1, 0, 0, 0, 24, 60, 870, 414, 4140, 1, 0, 0, 0, 0, -120, 240, -5922, 49, -21147, 1, 0, 0, 0, 0, 120, 360, -4830, 45416, -10088, 115975
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, ... -1, 1, 0, 0, 0, ... 2, -1, 2, 0, 0, ... -5, -2, -6, 6, 0, ... 15, 9, 24, -24, 24, ...
Links
- Seiichi Manyama, Antidiagonals n = 0..139, flattened
Crossrefs
Programs
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Ruby
def f(n) return 1 if n < 2 (1..n).inject(:*) end def ncr(n, r) return 1 if r == 0 (n - r + 1..n).inject(:*) / (1..r).inject(:*) end def A(k, n) ary = [1] (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]}} ary end def A292973(n) a = [] (0..n).each{|i| a << A(i, n - i)} ary = [] (0..n).each{|i| (0..i).each{|j| ary << a[i - j][j] } } ary end p A292973(20)
Formula
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