A145460 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where sequence a_k of column k is the exponential transform of C(n,k).

1, 1, 1, 1, 1, 2, 1, 0, 3, 5, 1, 0, 1, 10, 15, 1, 0, 0, 3, 41, 52, 1, 0, 0, 1, 9, 196, 203, 1, 0, 0, 0, 4, 40, 1057, 877, 1, 0, 0, 0, 1, 10, 210, 6322, 4140, 1, 0, 0, 0, 0, 5, 30, 1176, 41393, 21147, 1, 0, 0, 0, 0, 1, 15, 175, 7273, 293608, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 1176, 49932, 2237921, 678570

Offset: 0

Author

Alois P. Heinz, Oct 10 2008

Keywords

• nonn
• tabl

Comments

A(n,k) is also the number of ways of placing n labeled balls into indistinguishable boxes, where in each filled box k balls are seen at the top. E.g. A(3,1)=10:

|1.| |2.| |3.| |1|2| |1|2| |1|3| |1|3| |2|3| |2|3| |1|2|3|

|23| |13| |12| |3|.| |.|3| |2|.| |.|2| |1|.| |.|1| |.|.|.|

+--+ +--+ +--+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+ +-+-+-+

Examples

			Square array A(n,k) begins:
   1,   1,  1,  1,  1,  1,  ...
   1,   1,  0,  0,  0,  0,  ...
   2,   3,  1,  0,  0,  0,  ...
   5,  10,  3,  1,  0,  0,  ...
  15,  41,  9,  4,  1,  0,  ...
  52, 196, 40, 10,  5,  1,  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• N. J. A. Sloane, Transforms

Crossrefs

Columns k=0-9 give: A000110, A000248, A133189, A145453, A145454, A145455, A145456, A145457, A145458, A145459.

A(2n,n) gives A029651.

Cf.: A007318, A143398, A292948.

Programs

Maple

exptr:= proc(p) local g; g:=
          proc(n) option remember; `if`(n=0, 1,
             add(binomial(n-1, j-1) *p(j) *g(n-j), j=1..n))
        end: end:
A:= (n,k)-> exptr(i-> binomial(i, k))(n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

Exptr[p_] := Module[{g}, g[n_] := g[n] = If[n == 0, 1, Sum[Binomial[n-1, j-1] *p[j]*g[n-j], {j, 1, n}]]; g]; A[n_, k_] := Exptr[Function[i, Binomial[i, k]]][n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

Ruby

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 << (0..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ncr(j + 1, k) * ary[i - 1 - j]}}
  ary
end
def A145460(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 A145460(20) # Seiichi Manyama, Sep 28 2017

Formula

A(0,k) = 1 and A(n,k) = Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0. - Seiichi Manyama, Sep 28 2017