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

Author

Gus Wiseman, Feb 01 2018

Keywords

• nonn
• tabf

Comments

Row sums are 2^(n-1). First column is A008965. A regular version is A299070.

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

Crossrefs

Cf. A001045, A001221, A008965, A059966, A116608, A146289, A185700, A296373, A299070.

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