cp's OEIS Frontend

A316655 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

%I A316655 #9 Sep 14 2018 02:34:18
%S A316655 0,1,1,1,2,3,5,4,12,9,12,17,33,29,44,26,90,90,261,68,168,93,766,144,
%T A316655 197,307,575,269,2312,428,7068,236,625,1017,863,954,21965,3409,2342,
%U A316655 712
%N A316655 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.
%C A316655 A rooted tree is series-reduced if every non-leaf node has at least two branches.
%C A316655 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%F A316655 a(prime(n)) = A000669(n).
%F A316655 a(2^n) = A000311(n).
%e A316655 Sequence of sets of trees begins:
%e A316655 1:
%e A316655 2: 1
%e A316655 3: (11)
%e A316655 4: (12)
%e A316655 5: (1(11)), (111)
%e A316655 6: (1(12)), (2(11)), (112)
%e A316655 7: (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111)
%e A316655 8: (1(23)), (2(13)), (3(12)), (123)
%e A316655 9: (1(1(22))), (1(2(12))), (1(122)), (2(1(12))), (2(2(11))), (2(112)), ((11)(22)), ((12)(12)), (11(22)), (12(12)), (22(11)), (1122)
%t A316655 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A316655 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A316655 gro[m_]:=If[Length[m]==1,m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
%t A316655 Table[Length[gro[Flatten[MapIndexed[Table[#2,{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]],{n,20}]
%Y A316655 Cf. A000081, A000311, A000669, A001678, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660.
%Y A316655 Cf. A316651, A316652, A316653, A316654, A316656.
%K A316655 nonn,more
%O A316655 1,5
%A A316655 _Gus Wiseman_, Jul 09 2018
%E A316655 a(37)-a(40) from _Robert Price_, Sep 13 2018