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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).
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%I #39 Mar 15 2023 12:40:04

%S 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 0,0,0,24,120,1920,37633,1,0,0,0,0,120,1080,17640,394353,1,0,0,0,0,

%U 120,720,10080,183120,4596553,1,0,0,0,0,0,720,5040,100800,2116800,58941091

%N 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 Seiichi Manyama, <a href="/A293053/b293053.txt">Antidiagonals n = 0..139, flattened</a>

%F E.g.f. of column k: exp(x^(k+1)/(1-x)).

%F 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 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 Square array begins:

%e 1, 1, 1, 1, ...

%e 1, 0, 0, 0, ...

%e 3, 2, 0, 0, ...

%e 13, 6, 6, 0, ...

%e 73, 36, 24, 24, ...

%e 501, 240, 120, 120, ...

%p A:= proc(n, k) option remember; `if`(n=0, 1, add(

%p A(n-j, k)*binomial(n-1, j-1)*j!, j=1+k..n))

%p end:

%p seq(seq(A(n,d-n), n=0..d), d=0..12); # _Alois P. Heinz_, Sep 29 2017

%t 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 Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Nov 07 2020 *)

%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 << (k..i - 1).inject(0){|s, j| s + f(j + 1) * ncr(i - 1, j) * ary[i - 1 - j]}}

%o ary

%o end

%o def A293053(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 A293053(20)

%Y Columns k=0..3 give A000262, A052845, A293049, A293050.

%Y Rows n=0..1 give A000012, A000007.

%Y Main diagonal gives A000007.

%Y A(n,n-1) gives A000142(n).

%Y Cf. A293024, A293119.

%K nonn,tabl

%O 0,6

%A _Seiichi Manyama_, Sep 29 2017