OFFSET
1,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..449 (terms 1..200 from Alois P. Heinz)
M. Aguiar and A. Lauve, Antipode and Convolution Powers of the Identity in Graded Connected Hopf Algebras, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1083-1094.
M. Aguiar, A. Lauve, The characteristic polynomial of the Adams operators on graded connected Hopf algebras, arXiv:1403.7584 [math.RA], 2014.
FORMULA
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
EXAMPLE
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).
MAPLE
read transforms; EULERi([seq(n!, n=1..30)]);
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(factorial):
seq(a(n), n = 1..22); # Peter Luschny, Nov 21 2022
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Mike Zabrocki, Sep 05 2005
STATUS
approved