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A299027
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Number of compositions of n whose standard factorization into Lyndon words has all distinct weakly increasing factors.
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4
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1, 1, 3, 5, 11, 20, 38, 69, 125, 225, 400, 708, 1244, 2176, 3779, 6532, 11229, 19223, 32745, 55555, 93875, 158025, 265038, 443009, 738026, 1225649, 2029305, 3350167, 5515384, 9055678, 14830076, 24226115, 39480306, 64190026, 104130753, 168556588, 272268482
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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) = 11 compositions:
(5) = (5)
(41) = (4)*(1)
(14) = (14)
(32) = (3)*(2)
(23) = (23)
(131) = (13)*(1)
(113) = (113)
(212) = (2)*(12)
(122) = (122)
(1121) = (112)*(1)
(1112) = (1112)
Not included:
(311) = (3)*(1)*(1)
(221) = (2)*(2)*(1)
(2111) = (2)*(1)*(1)*(1)
(1211) = (12)*(1)*(1)
(11111) = (1)*(1)*(1)*(1)*(1)
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MATHEMATICA
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nn=50;
ser=Product[(1+x^n)^(PartitionsP[n]-DivisorSigma[0, n]+1), {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(vector(n, n, numbpart(n) - numdiv(n) + 1))} \\ Andrew Howroyd, Dec 01 2018
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CROSSREFS
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Cf. A001045, A032153, A034691, A049311, A059966, A098407, A116540, A185700, A270995, A283877, A292432, A293993, A296373, A299023, A299024, A299026.
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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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