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A330727 Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose degrees (atom multiplicities) are the prime indices of n.

%I A330727 #5 Jan 05 2020 08:11:03
%S A330727 1,1,1,1,2,1,3,2,1,3,1,7,7,1,5,5,1,5,9,5,1,9,11,1,9,28,36,16,1,10,24,
%T A330727 16,1,14,38,27,1,13,18,1,13,69,160,164,61,1,24,79,62,1,20,160,580,
%U A330727 1022,855,272,1,19,59,45,1,27,138,232,123,1,17,77,121,61
%N A330727 Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose degrees (atom multiplicities) are the prime indices of n.
%C A330727 A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.
%C A330727 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%F A330727 T(2^n,k) = A008826(n,k).
%e A330727 Triangle begins:
%e A330727    {}
%e A330727    1
%e A330727    1
%e A330727    1   1
%e A330727    1   2
%e A330727    1   3   2
%e A330727    1   3
%e A330727    1   7   7
%e A330727    1   5   5
%e A330727    1   5   9   5
%e A330727    1   9  11
%e A330727    1   9  28  36  16
%e A330727    1  10  24  16
%e A330727    1  14  38  27
%e A330727    1  13  18
%e A330727    1  13  69 160 164  61
%e A330727    1  24  79  62
%e A330727 For example, row n = 12 counts the following multisystems:
%e A330727   {1,1,2,3}  {{1},{1,2,3}}    {{{1}},{{1},{2,3}}}
%e A330727              {{1,1},{2,3}}    {{{1,1}},{{2},{3}}}
%e A330727              {{1,2},{1,3}}    {{{1}},{{2},{1,3}}}
%e A330727              {{2},{1,1,3}}    {{{1,2}},{{1},{3}}}
%e A330727              {{3},{1,1,2}}    {{{1}},{{3},{1,2}}}
%e A330727              {{1},{1},{2,3}}  {{{1,3}},{{1},{2}}}
%e A330727              {{1},{2},{1,3}}  {{{2}},{{1},{1,3}}}
%e A330727              {{1},{3},{1,2}}  {{{2}},{{3},{1,1}}}
%e A330727              {{2},{3},{1,1}}  {{{2,3}},{{1},{1}}}
%e A330727                               {{{3}},{{1},{1,2}}}
%e A330727                               {{{3}},{{2},{1,1}}}
%t A330727 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[Reverse[FactorInteger[n]],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A330727 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330727 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330727 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t A330727 Table[Length[Select[totm[nrmptn[n]],Depth[#]==k&]],{n,2,10},{k,2,Length[nrmptn[n]]}]
%Y A330727 Row sums are A318846.
%Y A330727 Final terms in each row are A330728.
%Y A330727 Row prime(n) is row n of A330784.
%Y A330727 Row 2^n is row n of A008826.
%Y A330727 Row n is row A181821(n) of A330667.
%Y A330727 Column k = 3 is A318284(n) - 2 for n > 2.
%Y A330727 Cf. A000111, A002846, A005121, A292504, A318812, A318813, A318847, A318848, A318849, A330475, A330666, A330935.
%K A330727 nonn,tabf
%O A330727 2,5
%A A330727 _Gus Wiseman_, Jan 04 2020