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A091160 Number of orbits of length n under the map whose periodic points are counted by A061687.

%I A091160 #18 Feb 19 2021 20:10:00
%S A091160 1,16,2835,2370752,6611343125,48887897438124,821067869874486556,
%T A091160 28006755051982013513984,1782755223314276717178818904,
%U A091160 198173677662343700104263938337400,36467946245662764068249155883368682252,10631160782054640951386529213624176084501136
%N A091160 Number of orbits of length n under the map whose periodic points are counted by A061687.
%C A091160 Old name was: A061687 appears to count the periodic points for a certain map. If so, then this is the sequence of the numbers of orbits of length n for that map.
%H A091160 Alois P. Heinz, <a href="/A091160/b091160.txt">Table of n, a(n) for n = 1..100</a>
%H A091160 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A091160 J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/SIXDENIERS/bell.html">Extended Bell and Stirling Numbers From Hypergeometric Exponentiation</a>, J. Integer Seqs. Vol. 4 (2001), #01.1.4.
%H A091160 Thomas Ward, <a href="http://web.archive.org/web/20081002082625/http://www.mth.uea.ac.uk/~h720/research/files/integersequences.html">Exactly realizable sequences</a>. <a href="/A091112/a091112.pdf">[local copy]</a>.
%F A091160 If b(n) is the (n+1)th term of A061687, then a(n) = (1/n)*Sum_{d|n} mu(d)*b(n/d).
%e A091160 b(1)=1, b(3)=8506, so a(3) = (1/3)*(8506-1) = 2835.
%p A091160 with(numtheory):
%p A091160 b:= proc(n) option remember;
%p A091160       `if`(n=0, 1, add(binomial(n, k)^6*(n-k)*b(k)/n, k=0..n-1))
%p A091160     end:
%p A091160 a:= n-> add(mobius(d)*b(n/d), d=divisors(n))/n:
%p A091160 seq(a(n), n=1..15);  # _Alois P. Heinz_, Mar 19 2014
%t A091160 b[n_] := b[n] = If[n==0, 1, Sum[Binomial[n, k]^6 (n-k)b[k]/n, {k, 0, n-1}]];
%t A091160 a[n_] := Sum[MoebiusMu[d] b[n/d], {d, Divisors[n]}]/n;
%t A091160 Array[a, 15] (* _Jean-François Alcover_, Nov 18 2020, after _Alois P. Heinz_ *)
%Y A091160 Cf. A061687.
%K A091160 nonn
%O A091160 1,2
%A A091160 _Thomas Ward_, Feb 24 2004
%E A091160 More terms from _Alois P. Heinz_, Mar 19 2014
%E A091160 Name clarified by _Michel Marcus_, May 13 2015