A054625 -id:A054625 - cp's OEIS frontend

A075195 Jablonski table T(n,k) read by antidiagonals: T(n,k) = number of necklaces with n beads of k colors.

1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 11, 6, 1, 6, 15, 24, 24, 8, 1, 7, 21, 45, 70, 51, 14, 1, 8, 28, 76, 165, 208, 130, 20, 1, 9, 36, 119, 336, 629, 700, 315, 36, 1, 10, 45, 176, 616, 1560, 2635, 2344, 834, 60, 1, 11, 55, 249, 1044, 3367, 7826, 11165, 8230, 2195, 108, 1

Offset: 1

Christian G. Bower, Sep 07 2002

From Richard L. Ollerton, May 07 2021: (Start)
Here, as in A000031, turning over is not allowed.
(1/n) * Dirichlet convolution of phi(n) and k^n. (End)

			The array T(n,k) for n >= 1, k >= 1 begins:
  1,  2,   3,    4,     5,     6,      7, ...
  1,  3,   6,   10,    15,    21,     28, ...
  1,  4,  11,   24,    45,    76,    119, ...
  1,  6,  24,   70,   165,   336,    616, ...
  1,  8,  51,  208,   629,  1560,   3367, ...
  1, 14, 130,  700,  2635,  7826,  19684, ...
  1, 20, 315, 2344, 11165, 39996, 117655, ...
From _Indranil Ghosh_, Mar 25 2017: (Start)
Triangle formed when the array is read by antidiagonals:
   1;
   2,  1;
   3,  3,   1;
   4,  6,   4,   1;
   5, 10,  11,   6,    1;
   6, 15,  24,  24,    8,    1;
   7, 21,  45,  70,   51,   14,    1;
   8, 28,  76, 165,  208,  130,   20,   1;
   9, 36, 119, 336,  629,  700,  315,  36,  1;
  10, 45, 176, 616, 1560, 2635, 2344, 834, 60, 1;
  ... (End)

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 86 (2.2.23).
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 496.
Louis Comtet, Analyse combinatoire, Tome 2, p. 104 #17, P.U.F., 1970.

Andrew Howroyd, Table of n, a(n) for n = 1..1275
E. Jablonski, Théorie des permutations et des arrangements complets, Journal de Liouville, 8 (1892), pp. 331-49.
E. Jablonski, Sur l'analyse combinatoire circulaire, Comptes-rendus de l'Académie des Sciences, April 11 1892, Paris.
E. Lucas, Sur les permutations circulaires avec répétition, Théorie des Nombres, Annexe VII, Gauthier-Villars, Paris, 1891. Mentions Moreau.
P. A. MacMahon, Application of a theory of permutations in circular procession to the theory of numbers, Proc. Lond. Math. Soc., 23 (1892), pp. 305-313. Mentions Jablonski, Lucas and Moreau.
Index entries for sequences related to necklaces

Columns 1-10: A000012, A000031, A001867, A001868, A001869, A054625-A054629.
Rows 1-10: A000027, A000217, A006527, A006528, A054620, A006565, A054621-A054624.
Main Diagonal: A056665. A054630 and A054631 are the upper and lower triangles.
Cf. A000010, A185651, A246935.

t[n_, k_] := (1/n)*Sum[EulerPhi[d]*k^(n/d), {d, Divisors[n]}]; Table[t[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 20 2014, after Philippe Deléham *)

T(n, k) = (1/n) * sumdiv(n, d, eulerphi(d)*k^(n/d));
for(n=1, 15, for(k=1, n, print1(T(k, n - k + 1),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

from sympy.ntheory import totient, divisors
def T(n,k): return sum(totient(d)*k**(n//d) for d in divisors(n))//n
for n in range(1, 16):
    print([T(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 25 2017

T(n,k) = (1/n)*Sum_{d | n} phi(d)*k^(n/d), where phi = Euler totient function A000010. - Philippe Deléham, Oct 08 2003
From Petros Hadjicostas, Feb 08 2021: (Start)
O.g.f. for column k >= 1: Sum_{n>=1} T(n,k)*x^n = -Sum_{j >= 1} (phi(j)/j) * log(1 - k*x^j).
Linear recurrence for row n >= 1: T(n,k) = Sum_{j=0..n} -binomial(j-n-1,j+1) * T(n,k-1-j) for k >= n + 2. (This recurrence is essentially due to Robert A. Russell, who contributed it in A321791.) (End)
From Richard L. Ollerton, May 07 2021: (Start)
T(n,k) = (1/n)*Sum_{i=1..n} k^gcd(n,i).
T(n,k) = (1/n)*Sum_{i=1..n} k^(n/gcd(n,i))*phi(gcd(n,i))/phi(n/gcd(n,i)).
T(n,k) = (1/n)*A185651(n,k) for n >= 1, k >= 1. (End)
Product_{n>=1} 1/(1 - x^n)^T(n,k) = Product_{n>=1} 1/(1 - k*x^n). - Seiichi Manyama, Apr 12 2025

Additional references from Philippe Deléham, Oct 08 2003

A056294 Number of n-bead necklace structures using a maximum of six different colored beads.

Original entry on oeis.org

1, 2, 3, 7, 12, 43, 126, 539, 2304, 11023, 54682, 284071, 1509852, 8195029, 45080666, 250641895, 1404374248, 7917211349, 44848645458, 255055231763, 1455247360128, 8326191290585, 47752990403134

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.

The second Mathematica program uses Gilbert and Riordan's recurrence formula, which they recommend for calculations. - Robert A. Russell, Feb 24 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
N. J. A. Sloane, Maple code for this and related sequences

Cf. A000013, A054625.

Adn[d_, n_] := Module[{ c, t1, t2}, t2 = 0; For[c = 1, c <= d, c++, If[Mod[d, c] == 0 , t2 = t2 + (x^c/c)*(E^(c*z) - 1)]]; t1 = E^t2; t1 = Series[t1, {z, 0, n+1}]; Coefficient[t1, z, n]*n!]; Pn[n_] := Module[{ d, e, t1}, t1 = 0; For[d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*Adn[d, n/d]/n]]; t1/(1 - x)]; Pnq[n_, q_] := Module[{t1}, t1 = Series[Pn[n], {x, 0, q+1}] ; Coefficient[t1, x, q]]; a[n_] := Pnq[n, 6]; Table[Print[an = a[n]]; an, {n, 1, 23}] (* Jean-François Alcover, Oct 04 2013, after N. J. A. Sloane's Maple code *)
Adn[d_, n_] := Adn[d, n] = If[1==n, DivisorSum[d, x^# &], Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n-1], x] x]];
Table[SeriesCoefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &] /(n (1 - x)), {x, 0, 6}], {n, 1, 40}] (* Robert A. Russell, Feb 24 2018 *)
From Robert A. Russell, May 29 2018: (Start)
Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 60], 6 StirlingS2[n/#+5, 6] - 90 StirlingS2[n/#+4, 6] + 510 StirlingS2[n/#+3, 6] - 1350 StirlingS2[n/#+2, 6] + 1644 StirlingS2[n/#+1, 6] - 720 StirlingS2[n/#, 6], Divisible[#, 30], 5 StirlingS2[n/#+5, 6] - 77 StirlingS2[n/#+4, 6] + 451 StirlingS2[n/#+3, 6] - 1243 StirlingS2[n/#+2, 6] + 1584 StirlingS2[n/#+1, 6] - 720 StirlingS2[n/#, 6], Divisible[#, 20], 4 StirlingS2[n/#+5, 6] - 62 StirlingS2[n/#+4, 6] + 364 StirlingS2[n/#+3, 6] - 998 StirlingS2[n/#+2, 6] + 1252 StirlingS2[n/#+1, 6] - 560 StirlingS2[n/#, 6], Divisible[#, 15], 3 StirlingS2[n/#+5, 6] - 48 StirlingS2[n/#+4, 6] + 291 StirlingS2[n/#+3, 6] - 825 StirlingS2[n/#+2, 6] + 1074 StirlingS2[n/#+1, 6] - 495 StirlingS2[n/#, 6], Divisible[#, 12], 5 StirlingS2[n/#+5, 6] - 76 StirlingS2[n/#+4, 6] + 439 StirlingS2[n/#+3, 6] - 1196 StirlingS2[n/#+2, 6] + 1524 StirlingS2[n/#+1, 6] - 720 StirlingS2[n/#, 6], Divisible[#, 10], 3 StirlingS2[n/#+5, 6] - 49 StirlingS2[n/#+4, 6] + 305 StirlingS2[n/#+3, 6] - 891 StirlingS2[n/#+2, 6] + 1192 StirlingS2[n/#+1, 6] - 560 StirlingS2[n/#, 6], Divisible[#, 6], 4 StirlingS2[n/#+5, 6] - 63 StirlingS2[n/#+4, 6] + 380 StirlingS2[n/#+3, 6] - 1089 StirlingS2[n/#+2, 6] + 1464 StirlingS2[n/#+1, 6] - 720 StirlingS2[n/#, 6], Divisible[#, 5], 2 StirlingS2[n/#+5, 6] - 33 StirlingS2[n/#+4, 6] + 209 StirlingS2[n/#+3, 6] - 629 StirlingS2[n/#+2, 6] + 886 StirlingS2[n/#+1, 6] - 455 StirlingS2[n/#, 6], Divisible[#, 4], 3 StirlingS2[n/#+5, 6] - 48 StirlingS2[n/#+4, 6] + 293 StirlingS2[n/#+3, 6] - 844 StirlingS2[n/#+2, 6] + 1132 StirlingS2[n/#+1, 6] - 560 StirlingS2[n/#, 6], Divisible[#, 3], 2 StirlingS2[n/#+5, 6] - 34 StirlingS2[n/#+4, 6] + 220 StirlingS2[n/#+3, 6] - 671 StirlingS2[n/#+2, 6] + 954 StirlingS2[n/#+1, 6] - 495 StirlingS2[n/#, 6], Divisible[#, 2], 2 StirlingS2[n/#+5, 6] - 35 StirlingS2[n/#+4, 6] + 234 StirlingS2[n/#+3, 6] - 737 StirlingS2[n/#+2, 6] + 1072 StirlingS2[n/#+1, 6] - 560 StirlingS2[n/#, 6], True, StirlingS2[n/#+5, 6] - 19 StirlingS2[n/#+4, 6] + 138 StirlingS2[n/#+3, 6] - 475 StirlingS2[n/#+2, 6] + 766 StirlingS2[n/#+1, 6] - 455 StirlingS2[n/#, 6]] &], {n, 1, 40}]
mx = 40; Drop[CoefficientList[Series[1-Sum[(EulerPhi[d] / d) Which[ Divisible[d, 60], Log[1-6x^d], Divisible[d, 30], (3 Log[1-6x^d] + Log[1-2x^d]) / 4, Divisible[d, 20], (5 Log[1-6x^d] + 2 Log[1-3x^d]) / 9, Divisible[d, 15], (5 Log[1-6x^d] + 3 Log[1-4x^d] + 3 Log[1-2x^d]) / 16, Divisible[d, 12], (4 Log[1-6x^d] + Log[1-x^d]) / 5, Divisible[d, 10], (11 Log[1-6x^d] + 8 Log[1-3x^d] + 9 Log[1-2x^d]) / 36, Divisible[d, 6], (11 Log[1-6x^d] + 5 Log[1-2x^d] + 4 Log[1-x^d]) / 20, Divisible[d, 5], (29 Log[1-6x^d] + 3 Log[1-4x^d] + 8 Log[1-3x^d] + 27 Log[1-2x^d] + 24 Log[1-x^d]) / 144, Divisible[d, 4], (16 Log[1-6x^d] + 10 Log[1-3x^d] + 9 Log[1-x^d]) / 45, Divisible[d, 3], (9 Log[1-6x^d] + 15 Log[1-4x^d] + 15 Log[1-2x^d] + 16 Log[1-x^d]) / 80, Divisible[d, 2], (19 Log[1-6x^d] + 40 Log[1-3x^d] + 45 Log[1-2x^d] + 36 Log[1-x^d]) / 180, True, (Log[1-6 x^d] + 15 Log[1-4 x^d] + 40 Log[1-3 x^d] + 135 Log[1-2 x^d] + 264 Log[1-x^d]) / 720], {d, 1, mx}], {x, 0, mx}], x], 1]
(End)

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.
From Robert A. Russell, May 29 2018: (Start)
a(n) = (1/n)*Sum_{d|n} phi(d) * ([d==0 mod 60] * (6*S2(n/d + 5, 6) - 90*S2(n/d + 4, 6) + 510*S2(n/d + 3, 6) - 1350*S2(n/d + 2, 6) + 1644*S2(n/d + 1, 6) - 720*S2(n/d, 6)) + [d==30 mod 60] * (5*S2(n/d + 5, 6) - 77*S2(n/d + 4, 6) + 451*S2(n/d + 3, 6) - 1243*S2(n/d + 2, 6) + 1584*S2(n/d + 1, 6) - 720*S2(n/d, 6)) + [d==20 mod 60 | d==40 mod 60] * (4*S2(n/d + 5, 6) - 62*S2(n/d + 4, 6) + 364*S2(n/d + 3, 6) - 998*S2(n/d + 2, 6) + 1252*S2(n/d + 1, 6) - 560*S2(n/d, 6)) + [d==15 mod 60 | d==45 mod 60] * (3*S2(n/d + 5, 6) - 48*S2(n/d + 4, 6) + 291*S2(n/d + 3, 6) - 825*S2(n/d + 2, 6) + 1074*S2(n/d + 1, 6) - 495*S2(n/d, 6)) + [d mod 60 in {12,24,36,48}] * (5*S2(n/d + 5, 6) - 76*S2(n/d + 4, 6) + 439*S2(n/d + 3, 6) - 1196*S2(n/d + 2, 6) + 1524*S2(n/d + 1, 6) - 720*S2(n/d, 6)) + [d=10 mod 60 | d==50 mod 60] * (3*S2(n/d +5 , 6) - 49*S2(n/d + 4, 6) + 305*S2(n/d + 3, 6) - 891*S2(n/d + 2, 6) + 1192*S2(n/d + 1, 6) - 560*S2(n/d, 6)) + [d mod 60 in {6,18,42,54}] * (4*S2(n/d + 5, 6) - 63*S2(n/d + 4, 6) + 380*S2(n/d + 3, 6) - 1089*S2(n/d + 2, 6) + 1464*S2(n/d + 1, 6) - 720*S2(n/d, 6)) + [d mod 60 in {5,25,35,55}] * (2*S2(n/d + 5, 6) - 33*S2(n/d + 4, 6) + 209*S2(n/d + 3, 6) - 629*S2(n/d + 2, 6) + 886*S2(n/d + 1, 6) - 455*S2(n/d, 6)) + [d mod 60 in {4,8,16,28,32,44,52,56}] * (3*S2(n/d + 5, 6) - 48*S2(n/d + 4, 6) + 293*S2(n/d + 3, 6) - 844*S2(n/d + 2, 6) + 1132*S2(n/d + 1, 6) - 560*S2(n/d, 6)) + [d mod 60 in {3,9,21,27,33,39,51,57}] * (2*S2(n/d + 5, 6) - 34*S2(n/d + 4, 6) + 220*S2(n/d + 3, 6) - 671*S2(n/d + 2, 6) + 954*S2(n/d + 1, 6) - 495*S2(n/d, 6)) + [d mod 60 in {2,14,22,26,34,38,46,58}] * (2*S2(n/d + 5, 6) - 35*S2(n/d + 4, 6) + 234*S2(n/d + 3, 6) - 737*S2(n/d + 2, 6) + 1072*S2(n/d + 1, 6) - 560*S2(n/d, 6)) + [d mod 60 in {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59}] *
  (S2[n/d + 5, 6) - 19*S2(n/d + 4, 6) + 138*S2(n/d + 3, 6) -
  475*S2(n/d + 2, 6) + 766*S2(n/d + 1,6) - 455*S2(n/d, 6))), where S2(n,k) is the Stirling subset number, A008277.
G.f.: 1 - Sum_{d>0} (phi(d) / d) * ([d==0 mod 60] * log(1-6x^d) + [d==30 mod 60] * (3*log(1-6x^d) + log(1-2x^d)) / 4 + [d==20 mod 60 | d==40 mod 60] * (5*log(1-6x^d) + 2*log(1-3x^d)) / 9 + [d==15 mod 60 | d==45 mod 60] * (5*log(1-6x^d) + 3*log(1-4x^d) + 3*log(1-2x^d)) / 16 + [d mod 60 in {12,24,36,48}] * (4*log(1-6x^d) + log(1-x^d)) / 5 + [d=10 mod 60 | d==50 mod 60] * (11*log(1-6x^d) + 8*log(1-3x^d) + 9*log(1-2x^d)) / 36 + [d mod 60 in {6,18,42,54}] * (11*log(1-6x^d) + 5*log(1-2x^d) + 4*log(1-x^d)) / 20 + [d mod 60 in {5,25,35,55}] * (29*log(1-6x^d) + 3*log(1-4x^d) + 8*log(1-3x^d) + 27*log(1-2x^d) + 24*log(1-x^d)) / 144 + [d mod 60 in {4,8,16,28,32,44,52,56}] * (16*log(1-6x^d) + 10*log(1-3x^d) + 9*log(1-x^d)) / 45 + [d mod 60 in {3,9,21,27,33,39,51,57}] * (9*log(1-6x^d) + 15*log(1-4x^d) + 15*log(1-2x^d) + 16*log(1-x^d)) / 80 + [d mod 60 in {2,14,22,26,34,38,46,58}] * (19*log(1-6x^d) + 40*log(1-3x^d) + 45*log(1-2x^d) + 36*log(1-x^d)) / 180 + [d mod 60 in {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59}] * (log(1-6x^d) + 15*log(1-4x^d) + 40*log(1-3x^d) + 135*log(1-2x^d) + 264*log(1-x^d)) / 720).
(End)

A056286 Number of n-bead necklaces with exactly six different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 2160, 23940, 211680, 1643544, 11748240, 79419180, 516257280, 3262443120, 20193277104, 123071707080, 741419995680, 4427490147480, 26264144909520, 155018841055596, 911509010154720, 5344538384445120, 31272099902089200, 182707081122261480

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed.

			For n=6, the 120 necklaces are A followed by the 120 permutations of BCDEF.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000031, A001867, A001868, A001869, A054625.

Column k=6 of A087854.

k=6; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

a(n) = A054625(n) - 6*A001869(n) + 15*A001868(n) - 20*A001867(n) + 15*A000031(n) - 6.
From Robert A. Russell, Sep 26 2018: (Start)
a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=6 is the number of colors and S2 is the Stirling subset number A008277.
G.f.: -Sum_{d>0} (phi(d)/d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=6 is the number of colors. (End)

A056341 Number of bracelets of length n using a maximum of six different colored beads.

Original entry on oeis.org

6, 21, 56, 231, 888, 4291, 20646, 107331, 563786, 3037314, 16514106, 90782986, 502474356, 2799220041, 15673673176, 88162676511, 497847963696, 2821127825971, 16035812864946, 91404068329560

Offset: 1

Marks R. Nester

nonn

Turning over will not create a new bracelet.

			For n=2, the 21 bracelets are AA, AB, AC, AD, AE, AF, BB, BC, BD, BE, BF, CC, CD, CE, CF, DD, DE, DF, EE, EF, and FF. - _Robert A. Russell_, Sep 24 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]

Cf. A000029, A054625.
Cf. a(n) = A081720(n,6), n >= 6. - Wolfdieter Lang, Jun 03 2012
Column 6 of A051137.
Equals (A054625 + A056488) / 2 = A054625 - A278642 = A278642 + A056488.

mx=40;CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-6*x^n]/n,{n,mx}]+(1+6 x+15 x^2)/(1-6 x^2))/2,{x,0,mx}],x] (* Herbert Kociemba, Nov 02 2016 *)
k=6; Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}] (* Robert A. Russell, Sep 24 2018 *)

a(n) = Sum_{d|n} phi(d)*6^(n/d)/(2*n);
a(n) = 6^((n+1)/2)/2 for n odd,
       (7/4)*6^(n/2) for n even.
G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 6*x^n)/n + (1+6*x+15*x^2)/(1-6*x^2))/2. - Herbert Kociemba, Nov 02 2016

A184291 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..5 arrays.

Original entry on oeis.org

6, 21, 21, 76, 351, 76, 336, 7826, 7826, 336, 1560, 210456, 1119936, 210456, 1560, 7826, 6047412, 181402676, 181402676, 6047412, 7826, 39996, 181410426, 31345666736, 176319685116, 31345666736, 181410426, 39996, 210126, 5597460306

Offset: 1

R. H. Hardin, Jan 10 2011

			Table starts
      6        21          76          336        1560          7826      39996
     21       351        7826       210456     6047412     181410426 5597460306
     76      7826     1119936    181402676 31345666736 5642220395616
    336    210456   181402676 176319685116
   1560   6047412 31345666736
   7826 181410426
  39996

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 31 terms from R. H. Hardin)
S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352v1 [math.CO], Jan 10, 2013.
S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.

Columns 1-3 are A054625, A184289, A184290.

Cf. A184271, A184284.

T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*6^(n*(k/LCM[c, d])), {d, Divisors[k]}], {c, Divisors[n]}]; Table[T[n-k+1, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 30 2017, after Andrew Howroyd *)

T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 6^(n*k/lcm(c,d)))); \\ Andrew Howroyd, Sep 27 2017

T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 6^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

A054613 a(n) = Sum_{d|n} phi(d)*6^(n/d).

Original entry on oeis.org

0, 6, 42, 228, 1344, 7800, 46956, 279972, 1681008, 10078164, 60474120, 362797116, 2176832112, 13060694088, 78364444284, 470185001040, 2821111589856, 16926659444832, 101559966840108, 609359740010604, 3656158500550080

Offset: 0

N. J. A. Sloane, Apr 16 2000

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1285

Column k=6 of A185651.

Cf. A054625.

a(n) = if(n==0, 0, sumdiv(n, d, eulerphi(d)*6^(n/d))); \\ Altug Alkan, Mar 16 2018

a(n) = n * A054625(n).

a(n) = Sum_{k=1..n} 6^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021

A121775 T(n, k) = Sum_{d|n} phi(n/d)*binomial(d,k) for n>0, T(0, 0) = 1. Triangle read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 5, 3, 1, 4, 8, 7, 4, 1, 5, 9, 10, 10, 5, 1, 6, 15, 20, 21, 15, 6, 1, 7, 13, 21, 35, 35, 21, 7, 1, 8, 20, 36, 60, 71, 56, 28, 8, 1, 9, 21, 42, 86, 126, 126, 84, 36, 9, 1, 10, 27, 59, 130, 215, 253, 210, 120, 45, 10, 1, 11, 21, 55, 165, 330, 462, 462, 330, 165, 55

Offset: 0

Paul D. Hanna, Aug 23 2006

For n>0, (1/n)*Sum_{k=0..n} T(n,k)*(c-1)^k is the number of n-bead necklaces with c colors. See the cross references.

			Triangle begins:
[ 0]  1;
[ 1]  1,  1;
[ 2]  2,  3,  1;
[ 3]  3,  5,  3,   1;
[ 4]  4,  8,  7,   4,   1;
[ 5]  5,  9, 10,  10,   5,   1;
[ 6]  6, 15, 20,  21,  15,   6,   1;
[ 7]  7, 13, 21,  35,  35,  21,   7,   1;
[ 8]  8, 20, 36,  60,  71,  56,  28,   8,  1;
[ 9]  9, 21, 42,  86, 126, 126,  84,  36,  9,  1;
[10] 10, 27, 59, 130, 215, 253, 210, 120, 45, 10, 1;

Cf. A053635 (row sums), A121776 (antidiagonal sums), A054630, A327029.

Cf. A000031 (c=2), A001867 (c=3), A001868 (c=4), A001869 (c=5), A054625 (c=6), A054626 (c=7), A054627 (c=8), A054628 (c=9), A054629 (c=10).

PARI
```
T(n,k)=if(n
				
```

# uses[DivisorTriangle from A327029]
DivisorTriangle(euler_phi, binomial, 13) # Peter Luschny, Aug 24 2019

A278642 Number of pairs of orientable necklaces with n beads and up to 6 colors; i.e., turning the necklace over does not leave it unchanged. The turned-over necklace is not included in the count.

Original entry on oeis.org

0, 0, 0, 20, 105, 672, 3535, 19350, 102795, 556010, 3010098, 16467450, 90619690, 502194420, 2798240265, 15671993560, 88156797855, 497837886000, 2821092554035, 16035752398770, 91403856697944, 522308167195260, 2991401733402075, 17168047238861070, 98716274117752900, 568605754068247644, 3280417827002225910, 18953525314104758810

Offset: 0

Herbert Kociemba, Nov 24 2016

nonn

Number of chiral bracelets of n beads using up to six different colors.

Column 6 of A293496.

Cf. A059076 (2 colors), A278639 (3 colors), A278640 (4 colors), A278641 (5 colors).

mx = 40; f[x_, k_] := (1 - Sum[EulerPhi[n] * Log[1 - k * x^n]/n,{n, mx}] - Sum[Binomial[k, i] * x^i, {i, 0, 2}]/(1 - k * x^2))/2; CoefficientList[Series[f[x, 6], {x, 0, mx}], x]
k = 6; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) - (k^Floor[(n + 1)/2] + k^Ceiling[(n + 1)/2])/4, {n, 30}], 0] (* Robert A. Russell, Sep 24 2018 *)

Equals (A054625(n) - A056488(n)) / 2 = A054625(n) - A056341(n) = A056341(n) - A056488(n), for n >= 1.
G.f.: k = 6, (1 - Sum_{n >= 1} phi(n)*log(1 - k*x^n)/n - Sum_{i = 0..2} Binomial[k, i]*x^i / ( 1 - k*x^2) )/2.
For n > 0, a(n) = -(k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/2n)* Sum_{d|n} phi(d)*k^(n/d), where k = 6 is the maximum number of colors. - Robert A. Russell, Sep 24 2018

cp's OEIS Frontend

A075195 Jablonski table T(n,k) read by antidiagonals: T(n,k) = number of necklaces with n beads of k colors.

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A056294 Number of n-bead necklace structures using a maximum of six different colored beads.

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A056286 Number of n-bead necklaces with exactly six different colored beads.

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A056341 Number of bracelets of length n using a maximum of six different colored beads.

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A184291 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..5 arrays.

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A054613 a(n) = Sum_{d|n} phi(d)*6^(n/d).

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A121775 T(n, k) = Sum_{d|n} phi(n/d)*binomial(d,k) for n>0, T(0, 0) = 1. Triangle read by rows, for 0 <= k <= n.

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