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Number of compositions of n whose standard factorization into Lyndon words has all strict compositions as factors.
4

%I #12 Dec 02 2018 03:20:50

%S 1,2,4,7,12,23,38,66,112,193,319,539,887,1466,2415,3951,6417,10428,

%T 16817,27072,43505,69560,110916,176469,279893,442742,698919,1100898,

%U 1729530,2712134,4244263,6628174,10332499,16077835,24972415,38729239,59958797,92685287

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

%H Andrew Howroyd, <a href="/A299023/b299023.txt">Table of n, a(n) for n = 1..500</a>

%F Euler transform of A032153.

%e The a(5) = 12 compositions:

%e (5) = (5)

%e (41) = (4)*(1)

%e (14) = (14)

%e (32) = (3)*(2)

%e (23) = (23)

%e (311) = (3)*(1)*(1)

%e (131) = (13)*(1)

%e (221) = (2)*(2)*(1)

%e (212) = (2)*(12)

%e (2111) = (2)*(1)*(1)*(1)

%e (1211) = (12)*(1)*(1)

%e (11111) = (1)*(1)*(1)*(1)*(1)

%e Not included:

%e (113) = (113)

%e (122) = (122)

%e (1121) = (112)*(1)

%e (1112) = (1112)

%t nn=50;

%t ser=Product[1/(1-x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{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(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 Cf. A001045, A032020, A032153, A034691, A049311, A059966, A089259, A098407, A116540, A185700, A270995, A296373, A299024, A299026, A299027.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 31 2018