A221650 Tetrahedron P(n,j,k) = T(j,k)*p(n-j), where T(j,k) = 1 if k divides j otherwise 0.

1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 3, 2, 2, 1, 0, 1, 1, 1, 0, 1, 5, 3, 3, 2, 0, 2, 1, 1, 0, 1, 1, 0, 0, 0, 1, 7, 5, 5, 3, 0, 3, 2, 2, 0, 2, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 11, 7, 7, 5, 0, 5, 3, 3, 0, 3, 2, 0, 0, 0, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Omar E. Pol, Jan 21 2013

Keywords

• nonn
• tabf

Comments

This tetrahedron shows a connection between divisors and partitions.

Conjecture 1: P(n,j,k) is the number of partitions of n that contain at least m parts of size k, where m = j/k, if k divides j otherwise P(n,j,k) = 0.

Conjecture 2: P(n,j,k) is the number of parts that are the m-th part of size k in all partitions of n, where m = j/k, if k divides j otherwise P(n,j,k) = 0.

The sum of all elements of slice n is A006128(n).

The sum of row j of slice n is A221530(n,j).

The sum of column k of slice n is A066633(n,k).

See also the tetrahedron of A221649.

Examples

			First six slices of tetrahedron are
---------------------------------------------------
n  j    k: 1  2  3  4  5  6      A221530   A006128
---------------------------------------------------
1  1       1,                       1         1
...................................................
2  1       1,                       1
2  2       1, 1,                    2         3
...................................................
3  1       2,                       2
3  2       1, 1,                    2
3  3       1, 0, 1,                 2         6
...................................................
4  1       3,                       3
4  2       2, 2,                    4
4  3       1, 0, 1,                 2
4  4       1, 1, 0, 1,              3        12
...................................................
5  1       5,                       5
5  2       3, 3,                    6
5  3       2, 0, 2,                 4
5  4       1, 1, 0, 1,              3
5  5       1, 0, 0, 0, 1,           2        20
...................................................
6  1       7,                       7
6  2       5, 5,                   10
6  3       3, 0, 3,                 6
6  4       2, 2, 0, 2,              6
6  5       1, 0, 0, 0, 1,           2
6  6       1, 1, 1, 0, 0, 1         4        35
...................................................
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11480 (rows n = 1..40 of the tetrahedron, flattened)

Crossrefs

Cf. A000005, A006128, A027750, A051731, A066633, A127093, A221530, A221649.

Programs

Mathematica

A221650row[n_]:=Flatten[Table[If[Divisible[j,k],PartitionsP[n-j],0],{j,n},{k,j}]];Array[A221650row,10] (* Paolo Xausa, Sep 26 2023 *)

Formula

P(n,j,k) = A051731(j,k)*A000041(n-j) = (1/k)*A221649(n,j,k).