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A032250 "DHK[ n ](2n)" (bracelet, identity, unlabeled, n parts, evaluated at 2n) transform of 1,1,1,1,...

%I A032250 #27 May 25 2019 02:06:57
%S A032250 1,1,1,2,10,29,113,368,1316,4490,15907,55866,199550,714601,2583575,
%T A032250 9385280,34311304,126018592,465044951,1722987050,6407739430,
%U A032250 23909854891,89493459541,335911158480,1264104712300
%N A032250 "DHK[ n ](2n)" (bracelet, identity, unlabeled, n parts, evaluated at 2n) transform of 1,1,1,1,...
%C A032250 From _Petros Hadjicostas_, Feb 24 2019: (Start)
%C A032250 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.
%C A032250 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.
%C A032250 We conjecture that gg(k, x)= Sum_{d|k} mu(d)*ff(k/d, x^d).
%C A032250 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].]
%C A032250 (End)
%H A032250 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H A032250 Petros Hadjicostas, <a href="/A032250/a032250.pdf">The aperiodic version of Herbert Kociemba's formula for bracelets with no reflection symmetry</a>, 2019.
%p A032250 # 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.
%p A032250 with(NumberTheory);
%p A032250 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;
%p A032250 gg := proc (k, x) add(Moebius(d)*ff(k/d, x^d), d in Divisors(k)); end proc;
%p A032250 vv := proc (n) simplify(subs(x = 0, diff(gg(n, x), x$(2*n)))/factorial(2*n)); end proc;
%p A032250 for i from 3 to 100 do print(i, vv(i)); end do; # _Petros Hadjicostas_, Feb 24 2019
%Y A032250 Cf. A008804, A032246, A032247, A032248, A032249.
%K A032250 nonn
%O A032250 1,4
%A A032250 _Christian G. Bower_