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A322784
Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.
6
1, 1, 4, 8, 29, 59, 311, 892, 4983, 21863, 126813, 678626, 4446565, 27644538, 195561593, 1384705697, 10613378402, 82864870101, 686673571479, 5832742205547, 51897707277698, 474889512098459, 4514467567213008, 44152005855085601, 446355422070799305, 4638590359349994120
OFFSET
0,3
COMMENTS
A multiset is uniform if all multiplicities are equal.
Also the number of factorizations into factors > 1 of primorial powers k in A100778 with sum of prime indices A056239(k) equal to n.
a(n) is the number of nonequivalent nonnegative integer matrices without zero rows or columns with equal column sums and total sum n up to permutation of rows. - Andrew Howroyd, Jan 11 2020
LINKS
FORMULA
a(n) = Sum_{d|n} A001055(A002110(n/d)^d).
a(n) = Sum_{d|n} A219727(n/d, d). - Andrew Howroyd, Jan 11 2020
EXAMPLE
The a(1) = 1 through a(4) = 29 multiset partitions:
{{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
{{1,2}} {{1,2,3}} {{1,1,2,2}}
{{1},{1}} {{1},{1,1}} {{1,2,3,4}}
{{1},{2}} {{1},{2,3}} {{1},{1,1,1}}
{{2},{1,3}} {{1,1},{1,1}}
{{3},{1,2}} {{1},{1,2,2}}
{{1},{1},{1}} {{1,1},{2,2}}
{{1},{2},{3}} {{1,2},{1,2}}
{{1},{2,3,4}}
{{1,2},{3,4}}
{{1,3},{2,4}}
{{1,4},{2,3}}
{{2},{1,1,2}}
{{2},{1,3,4}}
{{3},{1,2,4}}
{{4},{1,2,3}}
{{1},{1},{1,1}}
{{1},{1},{2,2}}
{{1},{2},{1,2}}
{{1},{2},{3,4}}
{{1},{3},{2,4}}
{{1},{4},{2,3}}
{{2},{2},{1,1}}
{{2},{3},{1,4}}
{{2},{4},{1,3}}
{{3},{4},{1,2}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
{{1},{2},{3},{4}}
MATHEMATICA
u[n_, k_]:=u[n, k]=If[n==1, 1, Sum[u[n/d, d], {d, Select[Rest[Divisors[n]], #<=k&]}]];
Table[Sum[u[Array[Prime, d, 1, Times]^(n/d), Array[Prime, d, 1, Times]^(n/d)], {d, Divisors[n]}], {n, 12}]
CROSSREFS
Row sums of A322786.
Sequence in context: A075308 A300461 A280085 * A297638 A256456 A270522
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 26 2018
EXTENSIONS
a(14)-a(15) from Alois P. Heinz, Jan 16 2019
Terms a(16) and beyond from Andrew Howroyd, Jan 11 2020
STATUS
approved