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

1, 8, 513, 115272, 70162625, 95640604266, 256797561193432, 1238094271228829120, 9993778343964199218438, 127849400250667505250954500, 2480163309080566931933236667234, 70354340598798824605743590305386600, 2830805474672999382519296750329811657242

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, Feb 24 2004

Extensions

More terms from Robert Israel, May 05 2015

Name clarified by M. F. Hasler, May 11 2015

Status

approved