This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A293496 #26 Sep 28 2018 23:34:51
%S A293496 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,4,3,0,0,0,0,10,15,12,1,0,0,0,20,45,
%T A293496 72,38,2,0,0,0,35,105,252,270,117,6,0,0,0,56,210,672,1130,1044,336,14,
%U A293496 0,0,0,84,378,1512,3535,5270,3795,976,30,0
%N A293496 Array read by antidiagonals: T(n,k) = number of chiral pairs of necklaces with n beads using a maximum of k colors.
%C A293496 An orientable necklace when turned over does not leave it unchanged. Only one necklace in each pair is included in the count.
%C A293496 The number of chiral bracelets. An achiral bracelet is the same as its reverse, while a chiral bracelet is equivalent to its reverse. - _Robert A. Russell_, Sep 28 2018
%H A293496 Andrew Howroyd, <a href="/A293496/b293496.txt">Table of n, a(n) for n = 1..1275</a>
%F A293496 T(n,k) = (A075195(n,k) - A284855(n,k)) / 2.
%F A293496 From _Robert A. Russell_, Sep 28 2018: (Start)
%F A293496 T(n, k) = -(k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 4 + (1/2n) * Sum_{d|n} phi(d) * k^(n/d)
%F A293496 G.f. for column k: -(kx/4)*(kx+x+2)/(1-kx^2) - Sum_{d>0} phi(d)*log(1-kx^d)/2d. (End)
%e A293496 Array begins:
%e A293496 ==========================================================
%e A293496 n\k | 1 2 3 4 5 6 7 8
%e A293496 ----+-----------------------------------------------------
%e A293496 1 | 0 0 0 0 0 0 0 0 ...
%e A293496 2 | 0 0 0 0 0 0 0 0 ...
%e A293496 3 | 0 0 1 4 10 20 35 56 ...
%e A293496 4 | 0 0 3 15 45 105 210 378 ...
%e A293496 5 | 0 0 12 72 252 672 1512 3024 ...
%e A293496 6 | 0 1 38 270 1130 3535 9156 20748 ...
%e A293496 7 | 0 2 117 1044 5270 19350 57627 147752 ...
%e A293496 8 | 0 6 336 3795 23520 102795 355656 1039626 ...
%e A293496 9 | 0 14 976 14060 106960 556010 2233504 7440216 ...
%e A293496 10 | 0 30 2724 51204 483756 3010098 14091000 53615016 ...
%e A293496 ...
%e A293496 For T(3,4)=4, the chiral pairs are ABC-ACB, ABD-ADB, ACD-ADC, and BCD-BDC.
%e A293496 For T(4,3)=3, the chiral pairs are AABC-AACB, ABBC-ACBB, and ABCC-ACCB. - _Robert A. Russell_, Sep 28 2018
%t A293496 b[n_, k_] := (1/n)*DivisorSum[n, EulerPhi[#]*k^(n/#) &];
%t A293496 c[n_, k_] := If[EvenQ[n], (k^(n/2) + k^(n/2 + 1))/2, k^((n + 1)/2)];
%t A293496 T[_, 1] = T[1, _] = 0; T[n_, k_] := (b[n, k] - c[n, k])/2;
%t A293496 Table[T[n - k + 1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Oct 11 2017, translated from PARI *)
%o A293496 (PARI) \\ here b(n,k) is A075195 and c(n,k) is A284855
%o A293496 b(n, k) = (1/n) * sumdiv(n, d, eulerphi(d)*k^(n/d));
%o A293496 c(n, k) = if(n % 2 == 0, (k^(n/2) + k^(n/2+1))/2, k^((n+1)/2));
%o A293496 T(n, k) = (b(n, k) - c(n, k)) / 2;
%Y A293496 Columns 2..6 are A059076, A278639, A278640, A278641, A278642.
%Y A293496 Cf. A075195, A081720, A284855.
%K A293496 nonn,tabl
%O A293496 1,18
%A A293496 _Andrew Howroyd_, Oct 10 2017