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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 (list; graph; refs; listen; history; text; internal format)
 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 Andrew Howroyd, Table of n, a(n) for n = 0..50 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. Cf. A001055, A005176, A056239, A072774, A100778, A219727, A295193, A306017 (unlabeled version), A319190, A319612, A322785, A322786, A322792. Sequence in context: A075308 A300461 A280085 * A297638 A256456 A270522 Adjacent sequences:  A322781 A322782 A322783 * A322785 A322786 A322787 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

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Last modified August 12 08:51 EDT 2020. Contains 336438 sequences. (Running on oeis4.)