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%I #51 May 29 2019 05:24:19
%S 4,10,28,56,113,197,340,544,856,1284,1896,2709,3816,5247,7128,9504,
%T 12540,16302,21001,26728,33748,42185,52364,64448,78832,95725,115600,
%U 138720,165648,196707,232560,273600,320601,374034,434796,503448,581020,668173,766084
%N "DHK[ 7 ]" (bracelet, identity, unlabeled, 7 parts) transform of 1,1,1,1,...
%C From _Petros Hadjicostas_, Feb 24 2019: (Start)
%C When k is odd >= 3, the DHK[k] transform of sequence c = (c(n): n >= 1), whose g.f. is C(x) = Sum_{n>=1} c(n)*x^n, has g.f. Sum_{n>=1} (DHK[k] c)_n*x^n = (1/2)*Sum_{d|k} mu(d)*((1/k)*C(x^d)^(k/d) - C(x^d)*C(x^(2*d))^((k/d) - 1)/2)).
%C For the current sequence we have k = 7 and c(n) = 1 for all n >= 1. Hence, C(x) = x/(1-x) and A(x) = Sum_{n>=1} a(n)*x^n = (x^k/2)*Sum_{d|k} mu(d)*((1/k)*(1-x^d)^(-k/d) - (1-x^d)^(-1)*(1-x^(2*d))^(-((k/d) - 1)/2)).
%C The latter g.f. agrees with _Herbert Kociemba_'s formula found in the documentations of sequences and A008804 and A032246 only when k is an odd prime. The reason is that (DHK[k] c)_n, with c=(1,1,1,...), is the number of aperiodic bracelets without reflection symmetry with k black beads and n-k white beads, while Herbert Kociemba's formula (in the documentations of sequences and A008804 and A032246) counts all (periodic and aperiodic) bracelets without reflection symmetry with k black beads and n-k white beads. Hence, in the case k is an odd prime, the two formulas agree.
%C When k is even, the g.f. of the DHK[k] transform of sequence c = (c(n): n >= 1) is much more complicated.
%C Note that Herbert Kociemba's formula for counting all (periodic and aperiodic) bracelets with no reflection symmetry is still valid even when k is even; e.g., see sequence A008804 for the case k=4. For k = 4, all bracelets with 4 black beads and n-k = n-4 white beads that have no reflection symmetry are aperiodic, but this is not true anymore for k even >= 6.
%C (End)
%H Colin Barker, <a href="/A032248/b032248.txt">Table of n, a(n) for n = 10..1000</a>
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H Petros Hadjicostas, <a href="/A032248/a032248.pdf">The aperiodic version of Herbert Kociemba's formula for bracelets with no reflection symmetry</a>, 2019.
%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-8,6,6,-8,1,0,-1,8,-6,-6,8,0,-3,1).
%F G.f.: x^7*(1/(14*(1 - x)^7) - 1/((2*(1 - x))*(1 - x^2)^3) + 3/(7*(1 - x^7))). - _Petros Hadjicostas_, Feb 24 2019
%F a(n) = 3*a(n-1) - 8*a(n-3) + 6*a(n-4) + 6*a(n-5) - 8*a(n-6) + a(n-7) - a(n-9) + 8*a(n-10) - 6*a(n-11) - 6*a(n-12) + 8*a(n-13) - 3*a(n-15) + a(n-16) for n>25. - _Colin Barker_, Feb 25 2019
%o (PARI) Vec(x^10*(4 - 2*x - 2*x^2 + 4*x^3 + x^4 - 2*x^5 + x^6) / ((1 - x)^7*(1 + x)^3*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)) + O(x^40)) \\ _Colin Barker_, Feb 25 2019
%Y Cf. A001399, A008804, A032246, A032247, A032250. Column k = 7 of A180472.
%K nonn,easy
%O 10,1
%A _Christian G. Bower_
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