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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
OFFSET
1,5
COMMENTS
For bracelets, chiral pairs are counted as one.
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
Row sums give A019537.
Cf. A087854 (oriented), A305540 (achiral), A305541 (chiral).
Sequence in context: A322266 A190284 A327639 * A034870 A324224 A373432
KEYWORD
nonn,tabl
AUTHOR
Marko Riedel, Jun 02 2016
STATUS
approved