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A221650
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Tetrahedron P(n,j,k) = T(j,k)*p(n-j), where T(j,k) = 1 if k divides j otherwise 0.
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10
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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
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OFFSET
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1,5
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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First six slices of tetrahedron are
---------------------------------------------------
---------------------------------------------------
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
...................................................
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MATHEMATICA
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A221650row[n_]:=Flatten[Table[If[Divisible[j, k], PartitionsP[n-j], 0], {j, n}, {k, j}]]; Array[A221650row, 10] (* Paolo Xausa, Sep 26 2023 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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