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.

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, 0, 0, 24, 140, 1057, 877, 1, 0, 0, 0, 24, 60, 870, 6322, 4140, 1, 0, 0, 0, 0, 120, 480, 5922, 41393, 21147, 1, 0, 0, 0, 0, 120, 360, 5250, 45416, 293608, 115975

Offset

0,6

Links

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

Formula

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

Example

Square array begins:
   1,  1,  1,  1,  1, ...
   1,  1,  0,  0,  0, ...
   2,  3,  2,  0,  0, ...
   5, 10,  6,  6,  0, ...
  15, 41, 24, 24, 24, ...

Prog

(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 << f(k) * (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A292978(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 A292978(20)

Crossrefs

Columns k=0-4 give: A000110, A000248, A216507, A292889, A292979.

Rows n=0 gives A000012.

Main diagonal gives A000142.

Cf. A292973.

Sequence in context: A088523 A222211 A145460 * A202178 A035543 A350548

Adjacent sequences: A292975 A292976 A292977 * A292979 A292980 A292981

Keyword

nonn,tabl

Author

Seiichi Manyama, Sep 27 2017

Status

approved