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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
 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-François 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: A322266 A190284 A327639 * A034870 A324224 A264622 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 April 7 19:49 EDT 2020. Contains 333306 sequences. (Running on oeis4.)