A343658 Array read by antidiagonals where A(n,k) is the number of ways to choose a multiset of k divisors of n.

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 2, 1, 1, 6, 5, 10, 3, 4, 1, 1, 7, 6, 15, 4, 10, 2, 1, 1, 8, 7, 21, 5, 20, 3, 4, 1, 1, 9, 8, 28, 6, 35, 4, 10, 3, 1, 1, 10, 9, 36, 7, 56, 5, 20, 6, 4, 1, 1, 11, 10, 45, 8, 84, 6, 35, 10, 10, 2, 1

Offset: 1

Author

Gus Wiseman, Apr 29 2021

Keywords

• nonn
• tabl

Comments

First differs from A343656 at A(4,2) = 6, A343656(4,2) = 5.

As a triangle, T(n,k) = number of ways to choose a multiset of n - k divisors of k.

Examples

			Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=1:  1   1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8   9
  n=3:  1   2   3   4   5   6   7   8   9
  n=4:  1   3   6  10  15  21  28  36  45
  n=5:  1   2   3   4   5   6   7   8   9
  n=6:  1   4  10  20  35  56  84 120 165
  n=7:  1   2   3   4   5   6   7   8   9
  n=8:  1   4  10  20  35  56  84 120 165
  n=9:  1   3   6  10  15  21  28  36  45
Triangle begins:
   1
   1   1
   1   2   1
   1   3   2   1
   1   4   3   3   1
   1   5   4   6   2   1
   1   6   5  10   3   4   1
   1   7   6  15   4  10   2   1
   1   8   7  21   5  20   3   4   1
   1   9   8  28   6  35   4  10   3   1
   1  10   9  36   7  56   5  20   6   4   1
   1  11  10  45   8  84   6  35  10  10   2   1
For example, row n = 6 counts the following multisets:
  {1,1,1,1,1}  {1,1,1,1}  {1,1,1}  {1,1}  {1}  {}
               {1,1,1,2}  {1,1,3}  {1,2}  {5}
               {1,1,2,2}  {1,3,3}  {1,4}
               {1,2,2,2}  {3,3,3}  {2,2}
               {2,2,2,2}           {2,4}
                                   {4,4}
Note that for n = 6, k = 4 in the triangle, the two multisets {1,4} and {2,2} represent the same divisor 4, so they are only counted once under A343656(4,2) = 5.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Crossrefs

Row k = 1 of the array is A000005.

Column n = 4 of the array is A000217.

Column n = 6 of the array is A000292.

Row k = 2 of the array is A184389.

The distinct products of these multisets are counted by A343656.

Antidiagonal sums of the array (or row sums of the triangle) are A343661.

A000312 = n^n.

A009998(n,k) = n^k (as an array, offset 1).

A007318 counts k-sets of elements of {1..n}.

A059481 counts k-multisets of elements of {1..n}.

Cf. A000169, A062319, A067824, A143773, A146291, A176029, A285572, A326077, A327527, A334996, A343652, A343657.

Programs

Mathematica

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,k],n-k],{n,10},{k,n}]

PARI

A(n,k) = binomial(numdiv(n) + k - 1, k)
{ for(n=1, 9, for(k=0, 8, print1(A(n,k), ", ")); print ) } \\ Andrew Howroyd, Jan 11 2024

Formula

A(n,k) = ((A000005(n), k)) = A007318(A000005(n) + k - 1, k).

T(n,k) = ((A000005(k), n - k)) = A007318(A000005(k) + n - k - 1, n - k).