A299072 Sequence is an irregular triangle read by rows with zeros removed where T(n,k) is the number of compositions of n whose standard factorization into Lyndon words has k distinct factors.
1, 2, 3, 1, 5, 3, 7, 9, 13, 17, 2, 19, 39, 6, 35, 72, 21, 59, 141, 55, 1, 107, 266, 132, 7, 187, 511, 300, 26, 351, 952, 660, 85, 631, 1827, 1395, 240, 3, 1181, 3459, 2901, 636, 15, 2191, 6595, 5977, 1554, 67, 4115, 12604, 12123, 3698, 228, 7711, 24173, 24504
Offset: 1
Examples
Triangle begins:
1
2
3 1
5 3
7 9
13 17 2
19 39 6
35 72 21
59 141 55 1
107 266 132 7
187 511 300 26
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1196
Programs
-
Mathematica
LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ]; qit[q_]:=If[#===Length[q],{q},Prepend[qit[Drop[q,#]],Take[q,#]]]&[Max@@Select[Range[Length[q]],LyndonQ[Take[q,#]]&]]; DeleteCases[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Union[qit[#]]]===k&]],{n,11},{k,n}],0,{2}] -
PARI
\\ here b(n) is A059966. b(n)={sumdiv(n, d, moebius(n/d) * (2^d-1))/n} A(n)=[Vecrev(p/y) | p<-Vec(prod(k=1, n, (1 - y + y/(1-x^k) + O(x*x^n))^b(k))-1)] my(T=A(15)); for(n=1, #T, print(T[n])) \\ Andrew Howroyd, Dec 08 2018
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