%I #90 Feb 07 2021 00:50:38
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,
%T 0,0,0,0,0,2,2,2,0,0,0,0,0,0,3,4,4,3,0,0,0,0,0,0,4,6,10,6,4,0,0,0,0,0,
%U 0,5,10,16,16,10,5,0,0,0,0,0,0,7,14,28,30,28,14,7,0,0,0,0,0,0,8,20,42,56,56,42,20,8,0,0,0,0,0,0,10,26,64,91,113,91,64,26,10,0,0,0,0,0,0,12,35,90,150,197,197,150,90,35,12,0,0,0,0,0,0,14,44,126,224,340,370,340,224,126,44,14,0,0,0,0,0,0,16,56,168,336,544,680,680,544,336,168,56,16,0,0,0
%N Triangle T(n, k) = OC(n, k; not -1), read by rows, where OC(n, k; not -1) is the number of k-subsets of Z_n without -1 as a multiplier, up to congruency.
%C Let Z_n = {0,1,...,n-1} denote the integers mod n.
%C Let S be a k-subset of Z_n.
%C Then S has multiplier -1 iff there is a z in Z_n for which S = -S + z. Otherwise, S doesn't have multiplier -1.
%C For example in Z_7 the set S = {0,1,2} has multiplier -1 since -S = {0,-1,-2} = {0,5,6} and then {0,1,2} = {0,5,6} + 2, so S = -S + 2. But S={0,1,3} doesn't have multiplier -1.
%C Let S and S' be two k-subsets of Z_n.
%C Define an equivalence relation on the set of k-subsets as follows: S is congruent to S' iff S=S'+z or S = -S' + z for some z in Z_n.
%C Then define OC(n, k) to be the number of such congruence classes.
%C And define OC(n, k; not -1) to be the number of such congruence classes in which the representative doesn't have -1 as a multiplier.
%C Then this sequence is the 'OC(n,k; not -1)' triangle read by rows.
%C For convenience we start the triangle at n = 0, and we have 0 <= k <= n.
%C See the McSorley and Schoen (2013) reference below for equivalent definitions of this sequence in terms of (n,k)-Ovals and k-compositions of n.
%C From _Petros Hadjicostas_, May 29 2019: (Start)
%C Here, T(n, k) is the number of bracelets (turnover necklaces) of length n that have no reflection symmetry and consist of k white beads and n - k black beads. (Bracelets that have no reflection symmetry are also known as chiral bracelets.)
%C It is also the number of dihedral compositions of n into k parts with no reflection symmetry. It is also the number of dihedral compositions of n into n - k parts with no reflection symmetry. (For a definition of a dihedral composition, see Knopfmacher and Robbins (2013) in the references.)
%C For MacMahon's method for transforming a cyclic composition into a necklace and vice versa, see the comments for sequence A308401. See also p. 273 in Sommerville (1909).
%C (End)
%H Andrew Howroyd, <a href="/A180472/b180472.txt">Table of n, a(n) for n = 0..1274</a>
%H Hansraj Gupta, <a href="https://web.archive.org/web/20200806162943/https://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a66_964.pdf">Enumeration of incongruent cyclic k-gons</a>, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.
%H Petros Hadjicostas, <a href="/A180472/a180472.pdf">The aperiodic version of Herbert Kociemba's formula for bracelets with no reflection symmetry</a>, 2019.
%H Arnold Knopfmacher and Neville Robbins, <a href="https://www.researchgate.net/publication/260006088_Some_Properties_of_Dihedral_Compositions">Some properties of dihedral compositions</a>, Util. Math. 92 (2013), 207-220.
%H John P. McSorley and Alan H. Schoen, <a href="http://dx.doi.org/10.1016/j.disc.2012.08.021">Rhombic tilings of (n,k)-Ovals, (n, k, lambda)-cyclic difference sets, and related topics</a>, Discrete Math. 313 (2013), 129-154. (See Table 1 on p. 137.) - From _N. J. A. Sloane_, Nov 26 2012
%H Frank Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H Vladimir S. Shevelev, <a href="http://www.math.bgu.ac.il/~shevelev/Shevelev_Neclaces.pdf">Necklaces and convex k-gons</a>, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
%H Vladimir S. Shevelev, <a href="https://web.archive.org/web/20200722171019/http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/2000c4e8_629.pdf">Necklaces and convex k-gons</a>, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
%H Duncan M. Y. Sommerville, <a href="https://doi.org/10.1112/plms/s2-7.1.263">On certain periodic properties of cyclic compositions of numbers</a>, Proc. London Math. Soc. S2-7(1) (1909), 263-313.
%F From _Petros Hadjicostas_, May 29 2019: (Start)
%F T(n,k) = -binomial(floor(n/2) - (k mod 2) * (1 - (n mod 2)), floor(k/2)) / 2 + Sum_{d|n, d|k} (phi(d)*binomial(n/d, k/d)) / (2*n) for n >= 1 and 0 <= k <= n. (This is a modification of formulas due to Gupta (1979), Shevelev (2004), and W. Bomfim in sequence A052307.)
%F T(n, k) = A052307(n, k) - A119963(n,k) for 0 <= k <= n. (See the comments in CROSSREFS by J. P. McSorley.)
%F T(n, k) = T(n, n - k) for 0 <= k <= n.
%F G.f. for column k >= 1: (x^k/2) * (-(1 + x)/(1 - x^2)^floor((k/2) + 1) + (1/k) * Sum_{m|k} phi(m)/(1 - x^m)^(k/m)). (This formula is due to Herbert Kociemba.)
%F (End)
%F Bivariate g.f.: Sum_{n,k >= 0} T(n, k)*x^n*y^k = (1/2) * (1 - (1 + x) * (1 + x*y) / (1 - x^2 * (1 + y^2)) - Sum_{d >= 1} (phi(d) / d) * log(1 - x^d * (1 + y^d))). - _Petros Hadjicostas_, Jun 15 2019
%e The triangle begins (with rows for n >= 0 and columns for k >= 0) as follows:
%e 0
%e 0 0
%e 0 0 0
%e 0 0 0 0
%e 0 0 0 0 0
%e 0 0 0 0 0 0
%e 0 0 0 1 0 0 0
%e 0 0 0 1 1 0 0 0
%e 0 0 0 2 2 2 0 0 0
%e 0 0 0 3 4 4 3 0 0 0
%e 0 0 0 4 6 10 6 4 0 0 0
%e 0 0 0 5 10 16 16 10 5 0 0 0
%e 0 0 0 7 14 28 30 28 14 7 0 0 0
%e 0 0 0 8 20 42 56 56 42 20 8 0 0 0
%e 0 0 0 10 26 64 91 113 91 64 26 10 0 0 0 0
%e ...
%e For example the row which corresponds to Z_7 is: 0 0 0 1 1 0 0 0.
%e The first '1' here corresponds to the 3-subsets of Z_7.
%e There are 4 congruence classes of the 3-subsets of Z_7, their representatives are {0,1,2}, {0,2,4}, {0,1,4} and {0,1,3}. The first 3 representatives have multiplier -1, but the last doesn't. Hence there is just one 3-subset of Z_7 without multiplier -1, up to congruency.
%o (PARI) T(n,k) = if ((n==0) && (k==0), 0, -binomial(floor(n/2) - (k % 2) * (1 - n % 2), floor(k/2)) / 2 + sumdiv(gcd(n,k), d, (eulerphi(d)*binomial(n/d, k/d))) / (2*n));tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, May 30 2019
%Y This sequence is A052307-A119963. The sequence A052307 is formed from the triangle whose (n, k)-term is the number of k-subsets of Z_n up to congruence, and the sequence A119963 is formed from the triangle whose (n, k)-term is the number of k-subsets of Z_n with multiplier -1 up to congruence.
%Y The row sums of the 'OC(n, k, not -1)' triangle above give sequence A059076.
%Y Cf. A001399 (column k = 3 with different offset), A008804 (column k = 4 with different offset), A032246 (column k = 5), A308401 (column k = 6), A032248 (column k = 7).
%K nonn,tabl
%O 0,40
%A _John P. McSorley_, Sep 06 2010
%E Name edited by _Petros Hadjicostas_, May 29 2019
%E Offset corrected by _Andrew Howroyd_, Sep 27 2019