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A299023 Number of compositions of n whose standard factorization into Lyndon words has all strict compositions as factors.

%I A299023 #12 Dec 02 2018 03:20:50
%S A299023 1,2,4,7,12,23,38,66,112,193,319,539,887,1466,2415,3951,6417,10428,
%T A299023 16817,27072,43505,69560,110916,176469,279893,442742,698919,1100898,
%U A299023 1729530,2712134,4244263,6628174,10332499,16077835,24972415,38729239,59958797,92685287
%N A299023 Number of compositions of n whose standard factorization into Lyndon words has all strict compositions as factors.
%H A299023 Andrew Howroyd, <a href="/A299023/b299023.txt">Table of n, a(n) for n = 1..500</a>
%F A299023 Euler transform of A032153.
%e A299023 The a(5) = 12 compositions:
%e A299023       (5) = (5)
%e A299023      (41) = (4)*(1)
%e A299023      (14) = (14)
%e A299023      (32) = (3)*(2)
%e A299023      (23) = (23)
%e A299023     (311) = (3)*(1)*(1)
%e A299023     (131) = (13)*(1)
%e A299023     (221) = (2)*(2)*(1)
%e A299023     (212) = (2)*(12)
%e A299023    (2111) = (2)*(1)*(1)*(1)
%e A299023    (1211) = (12)*(1)*(1)
%e A299023   (11111) = (1)*(1)*(1)*(1)*(1)
%e A299023 Not included:
%e A299023     (113) = (113)
%e A299023     (122) = (122)
%e A299023    (1121) = (112)*(1)
%e A299023    (1112) = (1112)
%t A299023 nn=50;
%t A299023 ser=Product[1/(1-x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}];
%t A299023 Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
%o A299023 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A299023 seq(N)={EulerT(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 A299023 Cf. A001045, A032020, A032153, A034691, A049311, A059966, A089259, A098407, A116540, A185700, A270995, A296373, A299024, A299026, A299027.
%K A299023 nonn
%O A299023 1,2
%A A299023 _Gus Wiseman_, Jan 31 2018