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 A326517 Number of normal multiset partitions of weight n where each part has a different size. 12
 1, 1, 2, 12, 28, 140, 956, 3520, 17792, 111600, 1144400, 4884064, 34907936, 214869920, 1881044032, 25687617152, 139175009920, 1098825972608, 8770328141888, 74286112885504, 784394159958848, 15114871659653952, 92392468773724544, 889380453354852416, 7652770202041529856 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A multiset partition is normal if it covers an initial interval of positive integers. LINKS Andrew Howroyd, Table of n, a(n) for n = 0..200 EXAMPLE The a(0) = 1 through a(3) = 12 normal multiset partitions:   {}  {{1}}  {{1,1}}  {{1,1,1}}              {{1,2}}  {{1,1,2}}                       {{1,2,2}}                       {{1,2,3}}                       {{1},{1,1}}                       {{1},{1,2}}                       {{1},{2,2}}                       {{1},{2,3}}                       {{2},{1,1}}                       {{2},{1,2}}                       {{2},{1,3}}                       {{3},{1,2}} 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_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]]; Table[Length[Select[Join@@mps/@allnorm[n], UnsameQ@@Length/@#&]], {n, 0, 6}] PROG (PARI) R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))} seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Feb 07 2020 CROSSREFS Row sums of A332253. Cf. A007837, A038041, A255906, A317583, A326026, A326514, A326518, A326519, A326520, A326521, A326533. Sequence in context: A034318 A338798 A345694 * A248119 A240764 A215784 Adjacent sequences:  A326514 A326515 A326516 * A326518 A326519 A326520 KEYWORD nonn AUTHOR Gus Wiseman, Jul 12 2019 EXTENSIONS Terms a(8) and beyond from Andrew Howroyd, Feb 07 2020 STATUS approved

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Last modified January 17 11:57 EST 2022. Contains 350394 sequences. (Running on oeis4.)