cp's OEIS Frontend

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.

%I A292978 #24 Jul 10 2022 09:41:15
%S A292978 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 A292978 0,0,24,140,1057,877,1,0,0,0,24,60,870,6322,4140,1,0,0,0,0,120,480,
%U A292978 5922,41393,21147,1,0,0,0,0,120,360,5250,45416,293608,115975
%N 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.
%H A292978 Seiichi Manyama, <a href="/A292978/b292978.txt">Antidiagonals n = 0..139, flattened</a>
%F A292978 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 A292978 Square array begins:
%e A292978    1,  1,  1,  1,  1, ...
%e A292978    1,  1,  0,  0,  0, ...
%e A292978    2,  3,  2,  0,  0, ...
%e A292978    5, 10,  6,  6,  0, ...
%e A292978   15, 41, 24, 24, 24, ...
%o A292978 (Ruby)
%o A292978 def f(n)
%o A292978   return 1 if n < 2
%o A292978   (1..n).inject(:*)
%o A292978 end
%o A292978 def ncr(n, r)
%o A292978   return 1 if r == 0
%o A292978   (n - r + 1..n).inject(:*) / (1..r).inject(:*)
%o A292978 end
%o A292978 def A(k, n)
%o A292978   ary = [1]
%o A292978   (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 A292978   ary
%o A292978 end
%o A292978 def A292978(n)
%o A292978   a = []
%o A292978   (0..n).each{|i| a << A(i, n - i)}
%o A292978   ary = []
%o A292978   (0..n).each{|i|
%o A292978     (0..i).each{|j|
%o A292978       ary << a[i - j][j]
%o A292978     }
%o A292978   }
%o A292978   ary
%o A292978 end
%o A292978 p A292978(20)
%Y A292978 Columns k=0-4 give: A000110, A000248, A216507, A292889, A292979.
%Y A292978 Rows n=0 gives A000012.
%Y A292978 Main diagonal gives A000142.
%Y A292978 Cf. A292973.
%K A292978 nonn,tabl
%O A292978 0,6
%A A292978 _Seiichi Manyama_, Sep 27 2017