login
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
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
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.
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
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