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A273891 Triangle read by rows: T(n,k) is the number of n-bead bracelets with exactly k different colored beads. 13
1, 1, 1, 1, 2, 1, 1, 4, 6, 3, 1, 6, 18, 24, 12, 1, 11, 56, 136, 150, 60, 1, 16, 147, 612, 1200, 1080, 360, 1, 28, 411, 2619, 7905, 11970, 8820, 2520, 1, 44, 1084, 10480, 46400, 105840, 129360, 80640, 20160, 1, 76, 2979, 41388, 255636, 821952, 1481760, 1512000, 816480, 181440 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

For bracelets, chiral pairs are counted as one.

LINKS

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

Marko Riedel, Maple code for A087854 and A273891.

FORMULA

T(n,k) = Sum_{i=0..k-1} (-1)^i * binomial(k,i) * A081720(n,k-i). - Andrew Howroyd, Mar 25 2017

From Robert A. Russell, Sep 26 2018: (Start)

T(n,k) = (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 S2 is the Stirling subset number A008277.

G.f. for column k>1: (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).

T(n,k) = (A087854(n,k) + A305540(n,k)) / 2 = A087854(n,k) - A305541(n,k) = A305541(n,k) + A305540(n,k).

(End)

EXAMPLE

Triangle begins with T(1,1):

1;

1,  1;

1,  2,    1;

1,  4,    6,     3;

1,  6,   18,    24,     12;

1, 11,   56,   136,    150,     60;

1, 16,  147,   612,   1200,   1080,     360;

1, 28,  411,  2619,   7905,  11970,    8820,    2520;

1, 44, 1084, 10480,  46400, 105840,  129360,   80640,  20160;

1, 76, 2979, 41388, 255636, 821952, 1481760, 1512000, 816480, 181440;

For T(4,2)=4, the arrangements are AAAB, AABB, ABAB, and ABBB, all achiral.

For T(4,4)=3, the arrangements are ABCD, ABDC, and ACBD, whose chiral partners are ADCB, ACDB, and ADBC respectively. - Robert A. Russell, Sep 26 2018

MATHEMATICA

(* t = A081720 *) 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}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Oct 07 2017, after Andrew Howroyd *)

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, 10}, {k, 1, n}] // Flatten (* Robert A. Russell, Sep 26 2018 *)

CROSSREFS

Columns 1-6: A057427, A056342, A056343, A056344, A056345, A056346.

Row sums give A019537.

Cf. A087854 (oriented), A305540 (achiral), A305541 (chiral).

Sequence in context: A255009 A156579 A190284 * A034870 A264622 A275017

Adjacent sequences:  A273888 A273889 A273890 * A273892 A273893 A273894

KEYWORD

nonn,tabl

AUTHOR

Marko Riedel, Jun 02 2016

STATUS

approved

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Last modified November 17 05:59 EST 2018. Contains 317275 sequences. (Running on oeis4.)