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 A091112 Number of orbits of length n under the map whose periodic points are counted by A061686. 7
 1, 8, 513, 115272, 70162625, 95640604266, 256797561193432, 1238094271228829120, 9993778343964199218438, 127849400250667505250954500, 2480163309080566931933236667234, 70354340598798824605743590305386600, 2830805474672999382519296750329811657242 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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". LINKS Robert Israel, Table of n, a(n) for n = 1..126 Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4. Thomas Ward, Exactly realizable sequences. [local copy]. FORMULA If b(n) is the (n+1)-th term of A061686, then a(n) = (1/n)*Sum_{d|n} mu(d) b(n/d). EXAMPLE b(1)=1, b(3)=1540, so a(3)=(1/3)(b(3)-b(1))=513. MAPLE a061686:= proc(n) option remember;   add(binomial(n, k)^5*(n-k)*procname(k)/n, k=0..n-1) end proc: a061686(0):= 1: a:= n -> 1/n * add(numtheory:-mobius(d)*a061686(n/d), d = numtheory:-divisors(n)): seq(a(n), n=1..6); # Robert Israel, May 05 2015 MATHEMATICA (* 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 *) PROG (PARI) A091112(n)=sumdiv(n, d, moebius(d)*A061686(n/d)) \\ M. F. Hasler, May 11 2015 CROSSREFS Cf. A061686. Sequence in context: A067505 A173058 A107672 * A015480 A159532 A003397 Adjacent sequences:  A091109 A091110 A091111 * A091113 A091114 A091115 KEYWORD nonn AUTHOR Thomas Ward (t.ward(AT)uea.ac.uk), Feb 24 2004 EXTENSIONS More terms from Robert Israel, May 05 2015 Name clarified by M. F. Hasler, May 11 2015 STATUS approved

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