|
|
A319001
|
|
Number of ordered multiset partitions of integer partitions of n where the sequence of GCDs of the partitions is weakly increasing.
|
|
6
|
|
|
1, 1, 3, 7, 18, 42, 105, 248, 606, 1450, 3507, 8415, 20305, 48785, 117502, 282574, 680137, 1636005, 3936841, 9470776, 22787529, 54822530, 131901491, 317336519, 763489051, 1836862947, 4419324581, 10632404189, 25580507505, 61543948594, 148068421107
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
If we form a multiorder by treating integer partitions (a,...,z) as multiarrows GCD(a, ..., z) <= {z, ..., a}, then a(n) is the number of triangles of weight n.
|
|
LINKS
|
|
|
EXAMPLE
|
The a(4) = 18 ordered multiset partitions:
{{4}} {{1,3}} {{2,2}} {{1,1,2}} {{1,1,1,1}}
{{1},{3}} {{2},{2}} {{1},{1,2}} {{1},{1,1,1}}
{{1,2},{1}} {{1,1,1},{1}}
{{1,1},{2}} {{1,1},{1,1}}
{{1},{1},{2}} {{1},{1},{1,1}}
{{1},{1,1},{1}}
{{1,1},{1},{1}}
{{1},{1},{1},{1}}
|
|
PROG
|
(PARI) \\ here B(n) is A000837 as vector.
B(n) = {dirmul(vector(n, k, moebius(k)), vector(n, k, numbpart(k)))}
seq(n) ={my(p=x*Ser(B(n))); Vec(1/prod(g=1, n, 1 - subst(p + O(x*x^(n\g)), x, x^g)))} \\ Andrew Howroyd, Jan 16 2023
|
|
CROSSREFS
|
Cf. A000837, A007716, A055887, A063834, A255397, A269134, A276024, A289508, A316222, A317545, A317546, A319002, A319003.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, Jan 16 2023
|
|
STATUS
|
approved
|
|
|
|