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A299023
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Number of compositions of n whose standard factorization into Lyndon words has all strict compositions as factors.
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4
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1, 2, 4, 7, 12, 23, 38, 66, 112, 193, 319, 539, 887, 1466, 2415, 3951, 6417, 10428, 16817, 27072, 43505, 69560, 110916, 176469, 279893, 442742, 698919, 1100898, 1729530, 2712134, 4244263, 6628174, 10332499, 16077835, 24972415, 38729239, 59958797, 92685287
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OFFSET
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1,2
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LINKS
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FORMULA
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EXAMPLE
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The a(5) = 12 compositions:
(5) = (5)
(41) = (4)*(1)
(14) = (14)
(32) = (3)*(2)
(23) = (23)
(311) = (3)*(1)*(1)
(131) = (13)*(1)
(221) = (2)*(2)*(1)
(212) = (2)*(12)
(2111) = (2)*(1)*(1)*(1)
(1211) = (12)*(1)*(1)
(11111) = (1)*(1)*(1)*(1)*(1)
Not included:
(113) = (113)
(122) = (122)
(1121) = (112)*(1)
(1112) = (1112)
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MATHEMATICA
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nn=50;
ser=Product[1/(1-x^n)^Total[(Length[#]-1)!&/@Select[IntegerPartitions[n], UnsameQ@@#&]], {n, nn}];
Table[SeriesCoefficient[ser, {x, 0, n}], {n, nn}]
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PROG
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(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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
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CROSSREFS
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Cf. A001045, A032020, A032153, A034691, A049311, A059966, A089259, A098407, A116540, A185700, A270995, A296373, A299024, A299026, A299027.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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