A326895 -id:A326895 - cp's OEIS frontend

A305541 Triangle read by rows: T(n,k) is the number of chiral pairs of color loops of length n with exactly k different colors.

0, 0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 12, 24, 12, 0, 1, 35, 124, 150, 60, 0, 2, 111, 588, 1200, 1080, 360, 0, 6, 318, 2487, 7845, 11970, 8820, 2520, 0, 14, 934, 10240, 46280, 105840, 129360, 80640, 20160, 0, 30, 2634, 40488, 254676, 821592, 1481760, 1512000, 816480, 181440, 0, 62, 7503, 158220, 1344900, 5873760, 14658840, 21772800, 19051200, 9072000, 1814400

Offset: 1

Robert A. Russell, Jun 04 2018

In other words, the number of n-bead bracelets with beads of exactly k different colors that when turned over are different from themselves. - Andrew Howroyd, Sep 13 2019

			Triangle T(n,k) begins:
  0;
  0,  0;
  0,  0,    1;
  0,  0,    3,     3;
  0,  0,   12,    24,     12;
  0,  1,   35,   124,    150,     60;
  0,  2,  111,   588,   1200,   1080,     360;
  0,  6,  318,  2487,   7845,  11970,    8820,    2520;
  0, 14,  934, 10240,  46280, 105840,  129360,   80640,  20160;
  0, 30, 2634, 40488, 254676, 821592, 1481760, 1512000, 816480, 181440;
  ...
For T(4,3)=3, the chiral pairs are AABC-AACB, ABBC-ACBB, and ABCC-ACCB.
For T(4,4)=3, the chiral pairs are ABCD-ADCB, ABDC-ACDB, and ACBD-ADBC.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Columns 2-6 are A059076, A305542, A305543, A305544, and A305545.
Row sums are A326895.
Cf. A087854, A273891, A293496, A305540, A309651.

Table[(k!/(2n)) DivisorSum[n, EulerPhi[#] StirlingS2[n/#, k] &] - (k!/4) (StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten

T(n,k) = {-k!*(stirling((n+1)\2,k,2) + stirling(n\2+1,k,2))/4 + k!*sumdiv(n,d, eulerphi(d)*stirling(n/d,k,2))/(2*n)} \\ Andrew Howroyd, Sep 13 2019

T(n,k) = -(k!/4)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/(2 n))*Sum_{d|n} phi(d)*S2(n/d,k), where S2(n,k) is the Stirling subset number A008277.
T(n,k) = A087854(n,k) - A273891(n,k).
T(n,k) = (A087854(n,k) - A305540(n,k)) / 2.
T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A293496(n, i). - Andrew Howroyd, Sep 13 2019

A326888 Number of length n asymmetric bracelets with integer entries that cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 1, 6, 48, 369, 3341, 33960, 393467, 5111052, 73753685, 1170468816, 20263758984, 380047813297, 7676106093000, 166114206886740, 3834434320842720, 94042629507381957, 2442147034668044933, 66942194905675830162, 1931543452344523775050, 58519191359155952404837

Offset: 1

Andrew Howroyd, Sep 12 2019

nonn

Asymmetric bracelets are those primitive (period n) bracelets that when turned over are different from themselves.

			For n = 4, the six asymmetric bracelets are  1123, 1223, 1233, 1234, 1243, 1324.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of A309651.

Cf. A309528.

\\ U(n,k) is A309528
U(n,k)={sumdiv(n, d, moebius(n/d)*(k^d/n - if(d%2, k^((d+1)/2), (k+1)*k^(d/2)/2)))/2}
a(n)={sum(k=1, n, U(n,k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))}

Moebius transform of A326895.

A327868 Number of achiral loops (necklaces or bracelets) of length n with integer entries that cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 3, 8, 13, 44, 75, 308, 541, 2612, 4683, 25988, 47293, 296564, 545835, 3816548, 7087261, 54667412, 102247563, 862440068, 1622632573, 14857100084, 28091567595, 277474957988, 526858348381, 5584100659412, 10641342970443, 120462266974148, 230283190977853

Offset: 0

Andrew Howroyd, Sep 28 2019

nonn

Achiral loops may also be called periodic palindromes.

			The a(4) = 8 achiral loops are:
  1111,
  1122, 1112, 1212, 1222,
  1213, 1232, 1323.
G.f. = 1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 13*x^5 + 44*x^6 + 75*x^7 + ... - _Michael Somos_, May 04 2022

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row sums of A305540.

Cf. A000670, A005649, A326895.

a[ n_] := If[n < 0, 0, Sum[ k!*(StirlingS2[Quotient[n+1, 2], k] + StirlingS2[Quotient[n+2, 2], k]), {k, 0, n+1}]/2]; (* Michael Somos, May 04 2022 *)
a[ n_] := If[n < 0, 0, With[{m = Quotient[n+1, 2]},
m!*SeriesCoefficient[1/(2 - Exp@x)^Mod[n, 2, 1], {x, 0, m}]]]; (* Michael Somos, May 04 2022 *)

a(n)={if(n<1, n==0, sum(k=0, n, k!*(stirling((n+1)\2, k, 2)+stirling(n\2+1, k, 2)))/2)}

a(n) = (1/2)*Sum_{k=0..n} k!*(Stirling2(floor((n+1)/2), k) + Stirling2(ceiling((n+1)/2), k)) for n > 0.

a(2n-1) = A000670(n), a(2n) = A005649(n). - Michael Somos, May 04 2022

cp's OEIS Frontend

A305541 Triangle read by rows: T(n,k) is the number of chiral pairs of color loops of length n with exactly k different colors.

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A326888 Number of length n asymmetric bracelets with integer entries that cover an initial interval of positive integers.

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A327868 Number of achiral loops (necklaces or bracelets) of length n with integer entries that cover an initial interval of positive integers.

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