A292978 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = k! * Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * T(n-1-i,k) for n > 0.
1, 1, 1, 1, 1, 2, 1, 0, 3, 5, 1, 0, 2, 10, 15, 1, 0, 0, 6, 41, 52, 1, 0, 0, 6, 24, 196, 203, 1, 0, 0, 0, 24, 140, 1057, 877, 1, 0, 0, 0, 24, 60, 870, 6322, 4140, 1, 0, 0, 0, 0, 120, 480, 5922, 41393, 21147, 1, 0, 0, 0, 0, 120, 360, 5250, 45416, 293608, 115975
Offset: 0
Examples
Square array begins: 1, 1, 1, 1, 1, ... 1, 1, 0, 0, 0, ... 2, 3, 2, 0, 0, ... 5, 10, 6, 6, 0, ... 15, 41, 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 << f(k) * (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}} ary end def A292978(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 A292978(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