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 A317583 Number of multiset partitions of normal multisets of size n such that all blocks have the same size. 16
 1, 4, 8, 30, 32, 342, 128, 3754, 11360, 56138, 2048, 3834670, 8192, 27528494, 577439424, 2681075210, 131072, 238060300946, 524288, 11045144602614, 115488471132032, 49840258213638, 8388608, 152185891301461434, 140102945910265344, 124260001149229146, 85092642310351607968 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A multiset is normal if it spans an initial interval of positive integers. a(n) is the number of nonnegative integer matrices with total sum n, nonzero rows and each column with the same sum with columns in nonincreasing lexicographic order. - Andrew Howroyd, Jan 15 2020 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 FORMULA a(p) = 2^p for prime p. - Andrew Howroyd, Sep 15 2018 a(n) = Sum_{d|n} A331315(n/d, d). - Andrew Howroyd, Jan 15 2020 EXAMPLE The a(3) = 8 multiset partitions:   {{1,1,1}}   {{1,1,2}}   {{1,2,2}}   {{1,2,3}}   {{1},{1},{1}}   {{1},{1},{2}}   {{1},{2},{2}}   {{1},{2},{3}} MATHEMATICA sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}]; mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]]; allnorm[n_]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]; Table[Length[Select[Join@@mps/@allnorm[n], SameQ@@Length/@#&]], {n, 8}] PROG (PARI) \\ here U(n, m) gives number for m blocks of size n. U(n, m)={sum(k=1, n*m, binomial(binomial(k+n-1, n)+m-1, m)*sum(r=k, n*m, binomial(r, k)*(-1)^(r-k)) )} a(n)={sumdiv(n, d, U(d, n/d))} \\ Andrew Howroyd, Sep 15 2018 CROSSREFS Cf. A038041, A255906, A298422, A306017, A306019, A306020, A306021, A320324, A322794, A326517, A326518, A326519, A326520, A326521, A331315. Sequence in context: A297638 A256456 A270522 * A020331 A248476 A082595 Adjacent sequences:  A317580 A317581 A317582 * A317584 A317585 A317586 KEYWORD nonn AUTHOR Gus Wiseman, Aug 01 2018 EXTENSIONS Terms a(9) and beyond from Andrew Howroyd, Sep 15 2018 STATUS approved

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Last modified January 22 01:28 EST 2022. Contains 350481 sequences. (Running on oeis4.)