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A221650 Tetrahedron P(n,j,k) = T(j,k)*p(n-j), where T(j,k) = 1 if k divides j otherwise 0. 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
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.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11480 (rows n = 1..40 of the tetrahedron, flattened)
FORMULA
P(n,j,k) = A051731(j,k)*A000041(n-j) = (1/k)*A221649(n,j,k).
EXAMPLE
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
...................................................
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 *)
CROSSREFS
Sequence in context: A158208 A348652 A117274 * A140883 A214021 A260516
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Jan 21 2013
STATUS
approved

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Last modified March 29 08:01 EDT 2024. Contains 371265 sequences. (Running on oeis4.)