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A091112 Number of orbits of length n under the map whose periodic points are counted by A061686.

%I A091112 #21 Feb 19 2021 20:10:00
%S A091112 1,8,513,115272,70162625,95640604266,256797561193432,
%T A091112 1238094271228829120,9993778343964199218438,
%U A091112 127849400250667505250954500,2480163309080566931933236667234,70354340598798824605743590305386600,2830805474672999382519296750329811657242
%N A091112 Number of orbits of length n under the map whose periodic points are counted by A061686.
%C A091112 Old Name was: "A061686 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 under that map".
%H A091112 Robert Israel, <a href="/A091112/b091112.txt">Table of n, a(n) for n = 1..126</a>
%H A091112 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 A091112 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 A091112 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 A091112 If b(n) is the (n+1)-th term of A061686, then a(n) = (1/n)*Sum_{d|n} mu(d) b(n/d).
%e A091112 b(1)=1, b(3)=1540, so a(3)=(1/3)(b(3)-b(1))=513.
%p A091112 a061686:= proc(n) option remember;
%p A091112   add(binomial(n,k)^5*(n-k)*procname(k)/n, k=0..n-1)
%p A091112 end proc:
%p A091112 a061686(0):= 1:
%p A091112 a:= n -> 1/n * add(numtheory:-mobius(d)*a061686(n/d), d = numtheory:-divisors(n)):
%p A091112 seq(a(n), n=1..6); # _Robert Israel_, May 05 2015
%t A091112 (* b = A061686 *) b[0]=1; b[n_] := b[n] = Sum[Binomial[n, k]^5*(n-k)*b[k]/ n, {k, 0, n-1}]; a[n_] := (1/n)*DivisorSum[n, MoebiusMu[#] * b[n/#] &]; Array[a, 20] (* _Jean-François Alcover_, Dec 04 2015 *)
%o A091112 (PARI) A091112(n)=sumdiv(n,d,moebius(d)*A061686(n/d)) \\ _M. F. Hasler_, May 11 2015
%Y A091112 Cf. A061686.
%K A091112 nonn
%O A091112 1,2
%A A091112 _Thomas Ward_, Feb 24 2004
%E A091112 More terms from _Robert Israel_, May 05 2015
%E A091112 Name clarified by _M. F. Hasler_, May 11 2015