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A293053 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(x^i).

%I A293053 #39 Mar 15 2023 12:40:04
%S A293053 1,1,1,1,0,3,1,0,2,13,1,0,0,6,73,1,0,0,6,36,501,1,0,0,0,24,240,4051,1,
%T A293053 0,0,0,24,120,1920,37633,1,0,0,0,0,120,1080,17640,394353,1,0,0,0,0,
%U A293053 120,720,10080,183120,4596553,1,0,0,0,0,0,720,5040,100800,2116800,58941091
%N A293053 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(x^i).
%H A293053 Seiichi Manyama, <a href="/A293053/b293053.txt">Antidiagonals n = 0..139, flattened</a>
%F A293053 E.g.f. of column k: exp(x^(k+1)/(1-x)).
%F A293053 A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = Sum_{i=k..n-1} (i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.
%F A293053 A(n,k) = 2*(n-1) * A(n-1,k) - (n-1)*(n-2) * A(n-2,k) + (k+1)!*binomial(n-1,k) * A(n-1-k,k) - k*(k+1)!*binomial(n-1,k+1) * A(n-2-k,k) for n > k+1. - _Seiichi Manyama_, Mar 15 2023
%e A293053 Square array begins:
%e A293053     1,   1,   1,   1, ...
%e A293053     1,   0,   0,   0, ...
%e A293053     3,   2,   0,   0, ...
%e A293053    13,   6,   6,   0, ...
%e A293053    73,  36,  24,  24, ...
%e A293053   501, 240, 120, 120, ...
%p A293053 A:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A293053       A(n-j, k)*binomial(n-1, j-1)*j!, j=1+k..n))
%p A293053     end:
%p A293053 seq(seq(A(n,d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Sep 29 2017
%t A293053 A[0, _] = 1; A[n_, k_] /; n <= k = 0; A[n_, k_] := A[n, k] = Sum[(i+1)! Binomial[n-1, i] A[n-1-i, k], {i, k, n-1}];
%t A293053 Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Nov 07 2020 *)
%o A293053 (Ruby)
%o A293053 def f(n)
%o A293053   return 1 if n < 2
%o A293053   (1..n).inject(:*)
%o A293053 end
%o A293053 def ncr(n, r)
%o A293053   return 1 if r == 0
%o A293053   (n - r + 1..n).inject(:*) / (1..r).inject(:*)
%o A293053 end
%o A293053 def A(k, n)
%o A293053   ary = [1]
%o A293053   (1..n).each{|i| ary << (k..i - 1).inject(0){|s, j| s + f(j + 1) * ncr(i - 1, j) * ary[i - 1 - j]}}
%o A293053   ary
%o A293053 end
%o A293053 def A293053(n)
%o A293053   a = []
%o A293053   (0..n).each{|i| a << A(i, n - i)}
%o A293053   ary = []
%o A293053   (0..n).each{|i|
%o A293053     (0..i).each{|j|
%o A293053       ary << a[i - j][j]
%o A293053     }
%o A293053   }
%o A293053   ary
%o A293053 end
%o A293053 p A293053(20)
%Y A293053 Columns k=0..3 give A000262, A052845, A293049, A293050.
%Y A293053 Rows n=0..1 give A000012, A000007.
%Y A293053 Main diagonal gives A000007.
%Y A293053 A(n,n-1) gives A000142(n).
%Y A293053 Cf. A293024, A293119.
%K A293053 nonn,tabl
%O A293053 0,6
%A A293053 _Seiichi Manyama_, Sep 29 2017