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A299024 Number of compositions of n whose standard factorization into Lyndon words has distinct strict compositions as factors.

%I A299024 #9 Dec 02 2018 03:21:11
%S A299024 1,1,3,4,7,13,21,34,58,98,158,258,421,676,1108,1777,2836,4544,7220,
%T A299024 11443,18215,28729,45203,71139,111518,174402,272367,424892,660563,
%U A299024 1025717,1590448,2460346,3800816,5862640,9026963,13885425,21321663,32695098,50073855
%N A299024 Number of compositions of n whose standard factorization into Lyndon words has distinct strict compositions as factors.
%H A299024 Andrew Howroyd, <a href="/A299024/b299024.txt">Table of n, a(n) for n = 1..500</a>
%F A299024 Weigh transform of A032153.
%e A299024 The a(5) = 7 compositions:
%e A299024       (5) = (5)
%e A299024      (41) = (4)*(1)
%e A299024      (14) = (14)
%e A299024      (32) = (3)*(2)
%e A299024      (23) = (23)
%e A299024     (131) = (13)*(1)
%e A299024     (212) = (2)*(12)
%e A299024 Not included:
%e A299024     (311) = (3)*(1)*(1)
%e A299024     (113) = (113)
%e A299024     (221) = (2)*(2)*(1)
%e A299024     (122) = (122)
%e A299024    (2111) = (2)*(1)*(1)*(1)
%e A299024    (1211) = (12)*(1)*(1)
%e A299024    (1121) = (112)*(1)
%e A299024    (1112) = (1112)
%e A299024   (11111) = (1)*(1)*(1)*(1)*(1)
%t A299024 nn=50;
%t A299024 ser=Product[(1+x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}];
%t A299024 Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
%o A299024 (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o A299024 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
%Y A299024 Cf. A001045, A032020, A032153, A034691, A049311, A050342, A059966, A089259, A098407, A116540, A185700, A270995, A283877,  A292432, A296373, A299023, A299026, A299027.
%K A299024 nonn
%O A299024 1,3
%A A299024 _Gus Wiseman_, Jan 31 2018