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A306017 Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.

%I A306017 #26 Feb 09 2020 14:32:37
%S A306017 1,1,4,6,17,14,66,30,189,222,550,112,4696,202,5612,30914,63219,594,
%T A306017 453125,980,3602695,5914580,1169348,2510,299083307,232988061,23248212,
%U A306017 2669116433,14829762423,9130,170677509317,13684,1724710753084,2199418340875,14184712185,38316098104262
%N A306017 Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.
%C A306017 A multiset partition of weight n is a finite multiset of finite nonempty multisets whose sizes sum to n.
%C A306017 Number of distinct nonnegative integer matrices with all row sums equal and total sum n up to row and column permutations. - _Andrew Howroyd_, Sep 05 2018
%C A306017 From _Gus Wiseman_, Oct 11 2018: (Start)
%C A306017 Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears the same number of times. For n = 4, non-isomorphic representatives of these 17 multiset partitions are:
%C A306017   {{1,1,1,1}}
%C A306017   {{1,1,2,2}}
%C A306017   {{1,2,3,4}}
%C A306017   {{1},{1,1,1}}
%C A306017   {{1},{1,2,2}}
%C A306017   {{1},{2,3,4}}
%C A306017   {{1,1},{1,1}}
%C A306017   {{1,1},{2,2}}
%C A306017   {{1,2},{1,2}}
%C A306017   {{1,2},{3,4}}
%C A306017   {{1},{1},{1,1}}
%C A306017   {{1},{1},{2,2}}
%C A306017   {{1},{2},{1,2}}
%C A306017   {{1},{2},{3,4}}
%C A306017   {{1},{1},{1},{1}}
%C A306017   {{1},{1},{2},{2}}
%C A306017   {{1},{2},{3},{4}}
%C A306017 (End)
%H A306017 Andrew Howroyd, <a href="/A306017/b306017.txt">Table of n, a(n) for n = 0..50</a>
%F A306017 For p prime, a(p) = 2*A000041(p).
%F A306017 a(n) = Sum_{d|n} A331485(n/d, d). - _Andrew Howroyd_, Feb 09 2020
%e A306017 Non-isomorphic representatives of the a(4) = 17 multiset partitions:
%e A306017   {{1,1,1,1}}
%e A306017   {{1,1,2,2}}
%e A306017   {{1,2,2,2}}
%e A306017   {{1,2,3,3}}
%e A306017   {{1,2,3,4}}
%e A306017   {{1,1},{1,1}}
%e A306017   {{1,1},{2,2}}
%e A306017   {{1,2},{1,2}}
%e A306017   {{1,2},{2,2}}
%e A306017   {{1,2},{3,3}}
%e A306017   {{1,2},{3,4}}
%e A306017   {{1,3},{2,3}}
%e A306017   {{1},{1},{1},{1}}
%e A306017   {{1},{1},{2},{2}}
%e A306017   {{1},{2},{2},{2}}
%e A306017   {{1},{2},{3},{3}}
%e A306017   {{1},{2},{3},{4}}
%t A306017 permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
%t A306017 K[q_List, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
%t A306017 RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
%t A306017 a[n_] := a[n] = If[n==0, 1, If[PrimeQ[n], 2 PartitionsP[n], Sum[ RowSumMats[ n/d, n, d], {d, Divisors[n]}]]];
%t A306017 Table[Print[n, " ", a[n]]; a[n], {n, 0, 35}] (* _Jean-François Alcover_, Nov 07 2019, after _Andrew Howroyd_ *)
%o A306017 (PARI) \\ See A318951 for RowSumMats.
%o A306017 a(n)={sumdiv(n,d,RowSumMats(n/d,n,d))} \\ _Andrew Howroyd_, Sep 05 2018
%Y A306017 Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306018, A306019, A306020, A306021, A318951.
%Y A306017 Cf. A064573, A279787, A305551, A319616, A319056, A331485.
%K A306017 nonn
%O A306017 0,3
%A A306017 _Gus Wiseman_, Jun 17 2018
%E A306017 Terms a(11) and beyond from _Andrew Howroyd_, Sep 05 2018