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A056346 Number of bracelets of length n using exactly six different colored beads. 3
0, 0, 0, 0, 0, 60, 1080, 11970, 105840, 821952, 5874480, 39713550, 258136200, 1631273220, 10096734312, 61536377700, 370710950400, 2213749658880, 13132080672480, 77509456944318, 455754569692680 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Turning over will not create a new bracelet.

REFERENCES

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]

LINKS

Table of n, a(n) for n=1..21.

FORMULA

a(n) = A056341(n) - 6*A032276(n) + 15*A032275(n) - 20*A027671(n) + 15*A000029(n) - 6.

From Robert A. Russell, Sep 27 2018: (Start)

a(n) = (k!/4) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/2n) * 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.: (k!/4) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2) - Sum_{d>0} (phi(d)/2d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=6 is the number of colors. (End)

EXAMPLE

For a(6)=60, pair up the 120 permutations of BCDEF, each with its reverse, such as BCDEF-FEDCB.  Precede the first of each pair with an A, such as ABCDEF.  These are the 60 arrangements, all chiral.  If we precede the second of each pair with an A, such as AFEDCB, we get the chiral partner of each. - Robert A. Russell, Sep 27 2018

MATHEMATICA

t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]);

T[n_, k_] := Sum[(-1)^i*Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];

a[n_] := T[n, 6];

Array[a, 21] (* Jean-Fran├žois Alcover, Nov 05 2017, after Andrew Howroyd *)

k=6; Table[k! DivisorSum[n, EulerPhi[#] StirlingS2[n/#, k]&]/(2n) + k!(StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k])/4, {n, 1, 30}] (* Robert A. Russell, Sep 27 2018 *)

PROG

(PARI) a(n) = my(k=6); (k!/4) * (stirling(floor((n+1)/2), k, 2) + stirling(ceil((n+1)/2), k, 2)) + (k!/(2*n))*sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2)); \\ Michel Marcus, Sep 29 2018

CROSSREFS

Column 6 of A273891.

Equals (A056286 + A056492) / 2 = A056286 - A305545 = A305545 + A056492.

Cf. A008277.

Sequence in context: A223348 A305545 A056352 * A283722 A105252 A269138

Adjacent sequences:  A056343 A056344 A056345 * A056347 A056348 A056349

KEYWORD

nonn

AUTHOR

Marks R. Nester

STATUS

approved

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Last modified May 27 21:52 EDT 2020. Contains 334671 sequences. (Running on oeis4.)