OFFSET
1,4
COMMENTS
From Petros Hadjicostas, Feb 24 2019: (Start)
Let ff(k, x) = x^k/2 * ( (1/k)*Sum_{n|k} phi(n)/(1 - x^n)^(k/n) - (1 + x)/(1 -x^2)^floor(k/2 + 1) ) be Herbert Kociemba's formula for the g.f. of the number of all bracelets with k black beads and n-k white beads that have no reflection symmetry.
Let gg(k, x) be the generating function of the number of all aperiodic bracelets with k black beads and n-k white beads that have no reflection symmetry.
We conjecture that gg(k, x)= Sum_{d|k} mu(d)*ff(k/d, x^d).
For n >= 3, a(n) is the coefficient of x^(2*n) of the Taylor expansion of gg(n, x) around x=0. [Bower has special definitions for DHK[1] and DHK[2].]
(End)
LINKS
C. G. Bower, Transforms (2)
Petros Hadjicostas, The aperiodic version of Herbert Kociemba's formula for bracelets with no reflection symmetry, 2019.
MAPLE
# This is a crude program that assumes the above conjecture is true (which was later proved in Hadjicostas (2019)). It is only valid for n >= 3 because of Bower's special definition of DHK[k] for the cases k=1 and k=2.
with(NumberTheory);
ff := proc (k, x) (1/2)*x^k*(add(phi(n)/(1-x^n)^(k/n), n in Divisors(k))/k-(x+1)/(1-x^2)^floor((1/2)*k+1)); end proc;
gg := proc (k, x) add(Moebius(d)*ff(k/d, x^d), d in Divisors(k)); end proc;
vv := proc (n) simplify(subs(x = 0, diff(gg(n, x), x$(2*n)))/factorial(2*n)); end proc;
for i from 3 to 100 do print(i, vv(i)); end do; # Petros Hadjicostas, Feb 24 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved