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A299024
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Number of compositions of n whose standard factorization into Lyndon words has distinct strict compositions as factors.
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4
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1, 1, 3, 4, 7, 13, 21, 34, 58, 98, 158, 258, 421, 676, 1108, 1777, 2836, 4544, 7220, 11443, 18215, 28729, 45203, 71139, 111518, 174402, 272367, 424892, 660563, 1025717, 1590448, 2460346, 3800816, 5862640, 9026963, 13885425, 21321663, 32695098, 50073855
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OFFSET
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1,3
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LINKS
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FORMULA
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EXAMPLE
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The a(5) = 7 compositions:
(5) = (5)
(41) = (4)*(1)
(14) = (14)
(32) = (3)*(2)
(23) = (23)
(131) = (13)*(1)
(212) = (2)*(12)
Not included:
(311) = (3)*(1)*(1)
(113) = (113)
(221) = (2)*(2)*(1)
(122) = (122)
(2111) = (2)*(1)*(1)*(1)
(1211) = (12)*(1)*(1)
(1121) = (112)*(1)
(1112) = (1112)
(11111) = (1)*(1)*(1)*(1)*(1)
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MATHEMATICA
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nn=50;
ser=Product[(1+x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n], UnsameQ@@#&]], {n, nn}];
Table[SeriesCoefficient[ser, {x, 0, n}], {n, nn}]
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PROG
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(PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
seq(N)={WeighT(Vec(sum(n=1, N-1, (n-1)!*x^(n*(n+1)/2)/prod(k=1, n, 1-x^k + O(x^N)))))} \\ Andrew Howroyd, Dec 01 2018
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CROSSREFS
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Cf. A001045, A032020, A032153, A034691, A049311, A050342, A059966, A089259, A098407, A116540, A185700, A270995, A283877, A292432, A296373, A299023, A299026, A299027.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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