This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382204 #28 Apr 05 2025 12:01:55
%S A382204 1,1,2,3,4,4,7,5,8,8,10,8,15,9,14,15,17,13,22,14,25,21,23,19,34,24,29,
%T A382204 28,37,27,45,29,44,38,43,43,59,40,51,48,69,48,71,52,73,69,72,61,93,72,
%U A382204 91,77,99,78,105,95,119,95,113,96,146,107,126,123,151,130
%N A382204 Number of normal multiset partitions of weight n into constant blocks with a common sum.
%C A382204 We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.
%H A382204 Christian Sievers, <a href="/A382204/b382204.txt">Table of n, a(n) for n = 0..25000</a>
%F A382204 G.f.: 1 + Sum_{s>=1} Sum_{k=1..A055874(s)} Product_{v=1..k} (1/(1-x^(s/v)) - 1). - _Christian Sievers_, Apr 05 2025
%e A382204 The a(1) = 1 through a(6) = 7 multiset partitions:
%e A382204 {1} {11} {111} {1111} {11111} {111111}
%e A382204 {1}{1} {2}{11} {11}{11} {2}{11}{11} {111}{111}
%e A382204 {1}{1}{1} {2}{2}{11} {2}{2}{2}{11} {22}{1111}
%e A382204 {1}{1}{1}{1} {1}{1}{1}{1}{1} {11}{11}{11}
%e A382204 {2}{2}{11}{11}
%e A382204 {2}{2}{2}{2}{11}
%e A382204 {1}{1}{1}{1}{1}{1}
%e A382204 The a(1) = 1 through a(7) = 5 factorizations:
%e A382204 2 4 8 16 32 64 128
%e A382204 2*2 3*4 4*4 3*4*4 8*8 3*4*4*4
%e A382204 2*2*2 3*3*4 3*3*3*4 9*16 3*3*3*4*4
%e A382204 2*2*2*2 2*2*2*2*2 4*4*4 3*3*3*3*3*4
%e A382204 3*3*4*4 2*2*2*2*2*2*2
%e A382204 3*3*3*3*4
%e A382204 2*2*2*2*2*2
%t A382204 allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A382204 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382204 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A382204 Table[Length[Join@@(Select[mps[#],SameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])],{n,0,5}]
%o A382204 (PARI) h(s,x)=my(t=0,p=1,k=1);while(s%k==0,p*=1/(1-x^(s/k))-1;t+=p;k+=1);t
%o A382204 lista(n)=Vec(1+sum(s=1,n,h(s,x+O(x*x^n)))) \\ _Christian Sievers_, Apr 05 2025
%Y A382204 Without a common sum we have A055887.
%Y A382204 Twice-partitions of this type are counted by A279789.
%Y A382204 Without constant blocks we have A326518.
%Y A382204 For distinct block-sums and strict blocks we have A381718.
%Y A382204 Factorizations of this type are counted by A381995.
%Y A382204 For distinct instead of equal block-sums we have A382203.
%Y A382204 For strict instead of constant blocks we have A382429.
%Y A382204 A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
%Y A382204 A001055 count multiset partitions of prime indices, strict A045778.
%Y A382204 A089259 counts set multipartitions of integer partitions.
%Y A382204 A255906 counts normal multiset partitions, row sums of A317532.
%Y A382204 A321469 counts multiset partitions with distinct block-sums, ranks A326535.
%Y A382204 Normal multiset partitions: A035310, A304969, A356945.
%Y A382204 Set multipartitions: A116540, A270995, A296119, A318360.
%Y A382204 Set multipartitions with distinct sums: A279785, A381806, A381870.
%Y A382204 Constant blocks with distinct sums: A381635, A381636, A381716.
%Y A382204 Cf. A007716, A034691, A034729, A116539, A255903, A317583, A326520, A382216.
%K A382204 nonn
%O A382204 0,3
%A A382204 _Gus Wiseman_, Mar 26 2025
%E A382204 Terms a(16) and beyond from _Christian Sievers_, Apr 04 2025