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%I #9 Dec 02 2018 03:21:47
%S 1,2,4,8,16,31,59,111,205,378,685,1238,2213,3940,6955,12221,21333,
%T 37074,64073,110267,188877,322244,547522,926903,1563370,2628008,
%U 4402927,7353656,12244434,20329271,33657560,55574996,91525882,150356718,246403694,402861907
%N Number of compositions of n whose standard factorization into Lyndon words has all weakly increasing factors.
%H Andrew Howroyd, <a href="/A299026/b299026.txt">Table of n, a(n) for n = 1..1000</a>
%F Euler transform of A167934.
%e The 2^6 - a(7) = 5 compositions of 7 whose Lyndon prime factors are not all weakly increasing: (11212), (1132), (1213), (1321), (142).
%t nn=50;
%t ser=Product[1/(1-x^n)^(PartitionsP[n]-DivisorSigma[0,n]+1),{n,nn}];
%t Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o seq(n)={EulerT(vector(n, n, numbpart(n) - numdiv(n) + 1))} \\ _Andrew Howroyd_, Dec 01 2018
%Y Cf. A001045, A032153, A034691, A049311, A059966, A098407, A167934, A185700, A270995, A296373, A299023, A299024, A299027.
%K nonn
%O 1,2
%A _Gus Wiseman_, Feb 01 2018