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 A296373 Triangle T(n,k) = number of compositions of n whose factorization into Lyndon words (aperiodic necklaces) is of length k. 16
 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 5, 3, 1, 1, 9, 12, 6, 3, 1, 1, 18, 21, 14, 6, 3, 1, 1, 30, 45, 27, 15, 6, 3, 1, 1, 56, 84, 61, 29, 15, 6, 3, 1, 1, 99, 170, 120, 67, 30, 15, 6, 3, 1, 1, 186, 323, 254, 136, 69, 30, 15, 6, 3, 1, 1, 335, 640, 510, 295, 142, 70, 30, 15, 6, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 Wikipedia, Lyndon word: Standard factorization FORMULA First column is A059966. EXAMPLE Triangle begins:     1;     1,   1;     2,   1,   1;     3,   3,   1,   1;     6,   5,   3,   1,   1;     9,  12,   6,   3,   1,   1;    18,  21,  14,   6,   3,   1,   1;    30,  45,  27,  15,   6,   3,   1,   1;    56,  84,  61,  29,  15,   6,   3,   1,   1;    99, 170, 120,  67,  30,  15,   6,   3,   1,   1;   186, 323, 254, 136,  69,  30,  15,   6,   3,   1,   1;   335, 640, 510, 295, 142,  70,  30,  15,   6,   3,   1,   1; MATHEMATICA neckQ[q_]:=Array[OrderedQ[{RotateRight[q, #], q}]&, Length[q]-1, 1, And]; aperQ[q_]:=UnsameQ@@Table[RotateRight[q, k], {k, Length[q]}]; qit[q_]:=If[#===Length[q], {q}, Prepend[qit[Drop[q, #]], Take[q, #]]]&[Max@@Select[Range[Length[q]], neckQ[Take[q, #]]&&aperQ[Take[q, #]]&]]; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Length[qit[#]]===k&]], {n, 12}, {k, n}] PROG (PARI) EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i ))-1)} A(n)=[Vecrev(p/y) | p<-EulerMT(y*vector(n, n, sumdiv(n, d, moebius(n/d) * (2^d-1))/n))] { my(T=A(12)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Dec 01 2018 CROSSREFS Cf. A000740, A001045, A008965, A019536, A059966, A060223, A185700, A228369, A232472, A277427, A281013, A296302, A296372. Sequence in context: A304942 A090011 A061554 * A088326 A216956 A285522 Adjacent sequences:  A296370 A296371 A296372 * A296374 A296375 A296376 KEYWORD nonn,tabl AUTHOR Gus Wiseman, Dec 11 2017 STATUS approved

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Last modified August 9 09:47 EDT 2020. Contains 336319 sequences. (Running on oeis4.)