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A321791 Table read by descending antidiagonals: T(n,k) is the number of unoriented cycles (bracelets) of length n using up to k available colors. 4
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, 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, 84, 231, 377, 430, 198, 30, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Table of n, a(n) for n=0..65.

Index entries for sequences related to bracelets

FORMULA

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)).

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

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

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.

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

EXAMPLE

Table begins with T(0,0):

  1 1  1    1     1      1       1        1        1         1         1 ...

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

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

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

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

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

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

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

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

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

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

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).

MATHEMATICA

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

CROSSREFS

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

Cf. A051137 (ascending antidiagonals).

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

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

Main diagonal gives A081721.

Sequence in context: A110555 A097805 A071919 * A339649 A339779 A277504

Adjacent sequences:  A321788 A321789 A321790 * A321792 A321793 A321794

KEYWORD

nonn,tabl,easy

AUTHOR

Robert A. Russell, Dec 18 2018

STATUS

approved

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Last modified February 26 11:15 EST 2021. Contains 341631 sequences. (Running on oeis4.)