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).

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, 0, 0, 0, 24, 120, 1920, 37633, 1, 0, 0, 0, 0, 120, 1080, 17640, 394353, 1, 0, 0, 0, 0, 120, 720, 10080, 183120, 4596553, 1, 0, 0, 0, 0, 0, 720, 5040, 100800, 2116800, 58941091

Offset: 0

Author

Seiichi Manyama, Sep 29 2017

Keywords

• nonn
• tabl

Examples

			Square array begins:
    1,   1,   1,   1, ...
    1,   0,   0,   0, ...
    3,   2,   0,   0, ...
   13,   6,   6,   0, ...
   73,  36,  24,  24, ...
  501, 240, 120, 120, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..139, flattened

Crossrefs

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

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

Main diagonal gives A000007.

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

Cf. A293024, A293119.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1)*j!, j=1+k..n))
    end:
seq(seq(A(n,d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 29 2017

Mathematica

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}];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Nov 07 2020 *)

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 << (k..i - 1).inject(0){|s, j| s + f(j + 1) * ncr(i - 1, j) * ary[i - 1 - j]}}
  ary
end
def A293053(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 A293053(20)

Formula

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

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.

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