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A299203 Number of enriched p-trees whose multiset of leaves is the integer partition with Heinz number n. 18

%I

%S 0,1,1,1,1,1,1,2,1,1,1,4,1,1,1,5,1,3,1,3,1,1,1,11,1,1,2,3,1,5,1,12,1,

%T 1,1,15,1,1,1,11,1,4,1,3,3,1,1,38,1,3,1,3,1,9,1,9,1,1,1,21,1,1,4,34,1,

%U 4,1,3,1,5,1,54,1,1,3,3,1,4,1,33,5,1,1,23,1,1,1,9,1,20,1,3,1,1,1,117,1,3,3,12,1,4,1,9,4,1,1,57,1,4,1,34

%N Number of enriched p-trees whose multiset of leaves is the integer partition with Heinz number n.

%C By convention, a(1) = 0.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%e a(54) = 9: (((22)2)1), ((222)1), (((22)1)2), (((21)2)2), ((221)2), ((22)(21)), ((22)21), ((21)22), (2221).

%e a(40) = 11: ((31)(11)), (((31)1)1), ((3(11))1), ((311)1), (3((11)1)), (3(111)), (((11)1)3), ((111)3), ((31)11), (3(11)1), (3111).

%e a(36) = 15: ((22)(11)), ((2(11))2), (((11)2)2), (((21)1)2), ((211)2), (((22)1)1), (((21)2)1), ((221)1), ((21)(21)), (22(11)), (2(11)2), ((11)22), ((22)11), ((21)21), (2211).

%t nn=120;

%t ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];

%t tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];

%t qci[y_]:=qci[y]=If[Length[y]===1,1,Sum[Times@@qci/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y]&]}]];

%t qci/@ptns

%Y Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299200, A299201, A299202.

%K nonn

%O 1,8

%A _Gus Wiseman_, Feb 05 2018

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Last modified February 20 20:31 EST 2020. Contains 332084 sequences. (Running on oeis4.)