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A112354
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Inverse Euler transform of n!. Also the number of sequences of permutations with no global descents which are Lyndon (smallest in lexicographic order of all cyclic shifts of the sequences) where the size of the sequence = sum of sizes of the permutations.
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6
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1, 1, 4, 17, 92, 572, 4156, 34159, 314368, 3199844, 35703996, 433421495, 5687955724, 80256874912, 1211781887796, 19496946534720, 333041104402860, 6019770246910128, 114794574818830716, 2303332661416242633, 48509766592884311132, 1069983257387132347080
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OFFSET
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1,3
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LINKS
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FORMULA
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Product_{k>=1} 1/(1-x^k)^{a(k)} = Sum_{n>=0} n! x^n.
a(n) ~ n! * (1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - 171/n^5 - 1542/n^6 - 16241/n^7 - 194973/n^8 - 2622610/n^9 - 39027573/n^10 - ...), for coefficients see A113869. - Vaclav Kotesovec, Sep 04 2014, extended Nov 27 2020
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EXAMPLE
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a(3) = 4 because (123), (213), (132) and (1,21) are all Lyndon.
a(4) = 17 because there are 13 permutations with no global descents of size 4 and (1,123), (1,213), (1,132) are all Lyndon.
a(5) = 92 = 71 permutations with no global descents+13 sequences of the form (1,pi) where pi in S_4 with no global descents+(1,1,1,21),(1,21,21),(1,1,123),(1,1,213),(1,1,132),(21,123),(21,213),(21,132).
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MAPLE
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read transforms; EULERi([seq(n!, n=1..30)]);
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(factorial):
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MATHEMATICA
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ff = Range[n = 22]!; s = {}; For[i = 1, i <= n, i++, AppendTo[s, i*ff[[i]] - Sum[s[[d]]*ff[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* Jean-François Alcover, Apr 15 2016 *)
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CROSSREFS
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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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