cp's OEIS Frontend

A316656 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

%I A316656 #5 Jul 09 2018 21:24:04
%S A316656 0,1,0,1,0,1,0,4,3,1,0,9,0,1,6,26,0,36,0,16,10,1,0,92,21,1,197,25,0,
%T A316656 100,0,236,15,1,53,474
%N A316656 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.
%C A316656 A rooted tree is series-reduced if every non-leaf node has at least two branches. It is an identity tree if no branch appears multiple times under the same root.
%C A316656 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%F A316656 a(prime(n>1)) = 0.
%F A316656 a(2^n) = A000311(n).
%e A316656 Sequence of sets of trees begins:
%e A316656    1:
%e A316656    2: 1
%e A316656    3:
%e A316656    4: (12)
%e A316656    5:
%e A316656    6: (1(12))
%e A316656    7:
%e A316656    8: (1(23)), (2(13)), (3(12)), (123)
%e A316656    9: (1(2(12))), (2(1(12))), (12(12))
%e A316656   10: (1(1(12)))
%e A316656   11:
%e A316656   12: (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), ((12)(13)), (12(13)), (13(12))
%t A316656 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A316656 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A316656 gro[m_]:=If[Length[m]==1,m,Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],UnsameQ@@#&]];
%t A316656 Table[Length[gro[Flatten[MapIndexed[Table[#2,{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]]],{n,30}]
%Y A316656 Cf. A000081, A000311, A000669, A001678, A004111, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660.
%Y A316656 Cf. A316651, A316652, A316653. A316654, A316655.
%K A316656 nonn,more
%O A316656 1,8
%A A316656 _Gus Wiseman_, Jul 09 2018