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%I #10 Dec 02 2018 03:22:29
%S 1,1,3,5,11,20,38,69,125,225,400,708,1244,2176,3779,6532,11229,19223,
%T 32745,55555,93875,158025,265038,443009,738026,1225649,2029305,
%U 3350167,5515384,9055678,14830076,24226115,39480306,64190026,104130753,168556588,272268482
%N Number of compositions of n whose standard factorization into Lyndon words has all distinct weakly increasing factors.
%H Andrew Howroyd, <a href="/A299027/b299027.txt">Table of n, a(n) for n = 1..1000</a>
%F Weigh transform of A167934.
%e The a(5) = 11 compositions:
%e (5) = (5)
%e (41) = (4)*(1)
%e (14) = (14)
%e (32) = (3)*(2)
%e (23) = (23)
%e (131) = (13)*(1)
%e (113) = (113)
%e (212) = (2)*(12)
%e (122) = (122)
%e (1121) = (112)*(1)
%e (1112) = (1112)
%e Not included:
%e (311) = (3)*(1)*(1)
%e (221) = (2)*(2)*(1)
%e (2111) = (2)*(1)*(1)*(1)
%e (1211) = (12)*(1)*(1)
%e (11111) = (1)*(1)*(1)*(1)*(1)
%t nn=50;
%t ser=Product[(1+x^n)^(PartitionsP[n]-DivisorSigma[0,n]+1),{n,nn}];
%t Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
%o (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o seq(n)={WeighT(vector(n, n, numbpart(n) - numdiv(n) + 1))} \\ _Andrew Howroyd_, Dec 01 2018
%Y Cf. A001045, A032153, A034691, A049311, A059966, A098407, A116540, A185700, A270995, A283877, A292432, A293993, A296373, A299023, A299024, A299026.
%K nonn
%O 1,3
%A _Gus Wiseman_, Feb 01 2018
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