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

1,2

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1196

Example

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

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}]

Prog

(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

Crossrefs

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

Sequence in context: A300518 A111609 A238344 * A169820 A334658 A352525

Adjacent sequences: A299069 A299070 A299071 * A299073 A299074 A299075

Keyword

nonn,tabf

Author

Gus Wiseman, Feb 01 2018

Status

approved