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

1, 16, 2835, 2370752, 6611343125, 48887897438124, 821067869874486556, 28006755051982013513984, 1782755223314276717178818904, 198173677662343700104263938337400, 36467946245662764068249155883368682252, 10631160782054640951386529213624176084501136

Offset: 1

Author

Thomas Ward, Feb 24 2004

Keywords

• nonn

Comments

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.

Examples

			b(1)=1, b(3)=8506, so a(3) = (1/3)*(8506-1) = 2835.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..100
• 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. A061687.

Programs

Maple

with(numtheory):
b:= proc(n) option remember;
      `if`(n=0, 1, add(binomial(n, k)^6*(n-k)*b(k)/n, k=0..n-1))
    end:
a:= n-> add(mobius(d)*b(n/d), d=divisors(n))/n:
seq(a(n), n=1..15);  # Alois P. Heinz, Mar 19 2014

Mathematica

b[n_] := b[n] = If[n==0, 1, Sum[Binomial[n, k]^6 (n-k)b[k]/n, {k, 0, n-1}]];
a[n_] := Sum[MoebiusMu[d] b[n/d], {d, Divisors[n]}]/n;
Array[a, 15] (* Jean-François Alcover, Nov 18 2020, after Alois P. Heinz *)

Formula

If b(n) is the (n+1)th term of A061687, then a(n) = (1/n)*Sum_{d|n} mu(d)*b(n/d).

Extensions

More terms from Alois P. Heinz, Mar 19 2014

Name clarified by Michel Marcus, May 13 2015