A234574 T(n,k) is the number of size k ordered submultisets of the regular multiset {1_1,1_2,...,1_(n-1),1_n, ... ,i_1,i_2,...,i_(n-1),i_n, ... ,n_1,n_2,...,n_(n-1),n_n} (which contains n copies of i for 1 <= i <= n).

1, 1, 1, 1, 2, 4, 6, 6, 1, 3, 9, 27, 78, 210, 510, 1050, 1680, 1680, 1, 4, 16, 64, 256, 1020, 4020, 15540, 58380, 210840, 722400, 2310000, 6745200, 17417400, 37837800, 63063000, 63063000, 1, 5, 25, 125, 625, 3125, 15620, 77980, 388220, 1923180, 9454620

Offset: 0

Author

Thomas Wieder, Dec 29 2013

Keywords

• nonn
• tabf

Comments

A181567 gives the case for unordered submultisets.

Examples

			For n=2 we have the regular multiset L = [1,1,2,2].
We get the following ordered submultisets from L:
For k=0 1 multiset: []
For k=1 2 multisets: [1], [2]
For k=2 4 multisets: [1,1], [1,2], [2,1], [2,2]
For k=3 6 multisets: [1,1,2], [1,2,1], [2,1,1], [1,2,2], [2,1,2], [2,2,1]
For k=4 6 multisets: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1].
Triangle begins with:
  1;
  1, 1;
  1, 2, 4, 6, 6;
  1, 3, 9, 27, 78, 210, 510, 1050, 1680, 1680;
  1, 4, 16, 64, 256, 1020, 4020, 15540, 58380, 210840, 722400, 2310000, 6745200, 17417400, 37837800, 63063000, 63063000;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..26, flattened
• Thomas Wieder, Maple program for A234574

Crossrefs

Cf. A181567.

Row sums give A274762.

Programs

Maple

# first Maple program: see link above
# second Maple program:
b:= proc(n, k, i) option remember; `if`(k=0, 1,
     `if`(i<1, 0, add(b(n, k-j, i-1)/j!, j=0..n)))
    end:
T:= (n, k)-> b(n, k, n)*k!:
seq(seq(T(n, k), k=0..n^2), n=0..5); # Alois P. Heinz, Jul 04 2016

Extensions

More terms from Alois P. Heinz, Jul 04 2016