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A296373
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Triangle T(n,k) = number of compositions of n whose factorization into Lyndon words (aperiodic necklaces) is of length k.
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16
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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
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OFFSET
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1,4
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LINKS
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FORMULA
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EXAMPLE
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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;
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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Cf. A000740, A001045, A008965, A019536, A059966, A060223, A185700, A228369, A232472, A277427, A281013, A296302, A296372.
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KEYWORD
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AUTHOR
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STATUS
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approved
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