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

Author

Thomas Ward, Feb 24 2004

Keywords

• nonn

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".

Examples

			b(1)=1, b(3)=1540, so a(3)=(1/3)(b(3)-b(1))=513.
		

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].

Crossrefs

Cf. A061686.

Programs

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 *)

PARI

A091112(n)=sumdiv(n,d,moebius(d)*A061686(n/d)) \\ M. F. Hasler, May 11 2015

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).

Extensions

More terms from Robert Israel, May 05 2015

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