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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. 14

%I

%S 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,

%T 1200,1080,360,1,28,411,2619,7905,11970,8820,2520,1,44,1084,10480,

%U 46400,105840,129360,80640,20160,1,76,2979,41388,255636,821952,1481760,1512000,816480,181440

%N Triangle read by rows: T(n,k) is the number of n-bead bracelets with exactly k different colored beads.

%C For bracelets, chiral pairs are counted as one.

%H Andrew Howroyd, <a href="/A273891/b273891.txt">Table of n, a(n) for n = 1..1275</a>

%H Marko Riedel, <a href="/A273891/a273891.maple.txt">Maple code for A087854 and A273891.</a>

%F T(n,k) = Sum_{i=0..k-1} (-1)^i * binomial(k,i) * A081720(n,k-i). - _Andrew Howroyd_, Mar 25 2017

%F From _Robert A. Russell_, Sep 26 2018: (Start)

%F 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.

%F 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).

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

%F (End)

%e Triangle begins with T(1,1):

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 4, 6, 3;

%e 1, 6, 18, 24, 12;

%e 1, 11, 56, 136, 150, 60;

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

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

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

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

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

%e 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

%t (* 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_ *)

%t 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 *)

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

%Y Row sums give A019537.

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

%K nonn,tabl

%O 1,5

%A _Marko Riedel_, Jun 02 2016

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Last modified May 28 10:32 EDT 2020. Contains 334681 sequences. (Running on oeis4.)