%I #39 Feb 08 2021 02:11:38

%S 1,1,0,1,1,0,1,2,1,0,1,3,3,1,0,1,4,6,4,1,0,1,5,10,10,6,1,0,1,6,15,20,

%T 21,8,1,0,1,7,21,35,55,39,13,1,0,1,8,28,56,120,136,92,18,1,0,1,9,36,

%U 84,231,377,430,198,30,1,0

%N Table read by descending antidiagonals: T(n,k) is the number of unoriented cycles (bracelets) of length n using up to k available colors.

%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>

%F T(n,k) = [n==0] + [n>0] * (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 4 + (1/(2*n)) * Sum_{d|n} phi(d) * k^(n/d)).

%F T(n,k) = (A075195(n,k) + A284855(n,k)) / 2.

%F T(n,k) = A075195(n,k) - A293496(n,k) = A293496(n,k) + A284855(n,k).

%F Linear recurrence for row n: T(n,k) = Sum_{j=0..n} -binomial(j-n-1,j+1) * T(n,k-1-j) for k >= n + 1.

%F O.g.f. for column k >= 0: Sum_{n>=0} T(n,k)*x^n = 3/4 + (1 + k*x)^2/(4*(1 - k*x^2)) - (1/2) * Sum_{d >= 1} (phi(d)/d) * log(1 - k*x^d). - _Petros Hadjicostas_, Feb 07 2021

%e Table begins with T(0,0):

%e 1 1 1 1 1 1 1 1 1 1 1 ...

%e 0 1 2 3 4 5 6 7 8 9 10 ...

%e 0 1 3 6 10 15 21 28 36 45 55 ...

%e 0 1 4 10 20 35 56 84 120 165 220 ...

%e 0 1 6 21 55 120 231 406 666 1035 1540 ...

%e 0 1 8 39 136 377 888 1855 3536 6273 10504 ...

%e 0 1 13 92 430 1505 4291 10528 23052 46185 86185 ...

%e 0 1 18 198 1300 5895 20646 60028 151848 344925 719290 ...

%e 0 1 30 498 4435 25395 107331 365260 1058058 2707245 6278140 ...

%e 0 1 46 1219 15084 110085 563786 2250311 7472984 21552969 55605670 ...

%e 0 1 78 3210 53764 493131 3037314 14158228 53762472 174489813 500280022 ...

%e For T(3,3)=10, the unoriented cycles are 9 achiral (AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, CCC) and one chiral pair (ABC-ACB).

%t Table[If[k>0, DivisorSum[k, EulerPhi[#](n-k)^(k/#)&]/(2k) + ((n-k)^Floor[(k+1)/2]+(n-k)^Ceiling[(k+1)/2])/4, 1], {n, 0, 12}, {k, 0, n}] // Flatten

%Y Cf. A075195 (oriented), A293496(chiral), A284855 (achiral).

%Y Cf. A051137 (ascending antidiagonals).

%Y Columns 0-6 are A000007, A000012, A000029, A027671, A032275, A032276, and A056341.

%Y Rows 0-7 are A000012, A001477, A000217, A000292, A002817, A060446, A027670, and A060532.

%Y Main diagonal gives A081721.

%K nonn,tabl,easy

%O 0,8

%A _Robert A. Russell_, Dec 18 2018