A294018 Number of strict trees whose leaves are the parts of the integer partition with Heinz number n.

0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 3, 1, 0, 1, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 0, 1, 4, 1, 1, 1, 3, 1, 6, 1, 1, 1, 1, 1, 4, 1, 1, 0, 1, 1, 8, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 4, 1, 1, 6, 1, 4, 1, 1, 1, 4, 1, 1, 1, 1, 1, 13

Offset: 1

Author

Gus Wiseman, Feb 06 2018

Keywords

• nonn

Comments

By convention a(1) = 0.

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

Examples

			The a(84) = 8 strict trees: (((42)1)1), (((41)2)1), ((4(21))1), ((421)1), (((41)1)2), ((41)(21)), ((41)21), (4(21)1).
		

Crossrefs

Cf. A000009, A000041, A000720, A001222, A056239, A063834, A196545, A215366, A273873, A281145, A289501, A296150, A299201, A299202, A299203.

Programs

Mathematica

nn=120;
ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
qci[y_]:=qci[y]=If[Length[y]===1,1,Sum[Times@@qci/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y,UnsameQ@@Total/@#]&]}]];
qci/@ptns

Formula

A273873(n) = Sum_{i=1..A000041(n)} a(A215366(n,i)).