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%I #24 Jul 10 2022 09:41:15
%S 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,
%T 0,0,24,140,1057,877,1,0,0,0,24,60,870,6322,4140,1,0,0,0,0,120,480,
%U 5922,41393,21147,1,0,0,0,0,120,360,5250,45416,293608,115975
%N 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.
%H Seiichi Manyama, <a href="/A292978/b292978.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, 3, 2, 0, 0, ...
%e 5, 10, 6, 6, 0, ...
%e 15, 41, 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 << f(k) * (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
%o ary
%o end
%o def A292978(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 A292978(20)
%Y Columns k=0-4 give: A000110, A000248, A216507, A292889, A292979.
%Y Rows n=0 gives A000012.
%Y Main diagonal gives A000142.
%Y Cf. A292973.
%K nonn,tabl
%O 0,6
%A _Seiichi Manyama_, Sep 27 2017
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