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A276550 Array read by antidiagonals: T(n,k) = number of primitive (period n) bracelets using a maximum of k different colored beads.

%I A276550 #30 Mar 30 2023 06:14:48
%S A276550 1,2,0,3,1,0,4,3,2,0,5,6,7,3,0,6,10,16,15,6,0,7,15,30,45,36,8,0,8,21,
%T A276550 50,105,132,79,16,0,9,28,77,210,372,404,195,24,0,10,36,112,378,882,
%U A276550 1460,1296,477,42,0,11,45,156,630,1848,4220,5890,4380,1209,69,0
%N A276550 Array read by antidiagonals: T(n,k) = number of primitive (period n) bracelets using a maximum of k different colored beads.
%C A276550 Turning over will not create a new bracelet.
%H A276550 Andrew Howroyd, <a href="/A276550/b276550.txt">Table of n, a(n) for n = 1..1275</a>
%H A276550 G. Melançon, C. Reutenauer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Reutenauer/reut3.html">On a Class of Lyndon Words Extending Christoffel Words and Related to a Multidimensional Continued Fraction Algorithm</a>, J. Int. Seq. 16 (2013) #13.9.7, Corollary 6.
%H A276550 F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A276550 F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]
%F A276550 T(n, k) = Sum_{d|n} mu(n/d) * A081720(d,k) for k<=n. Corrected Jan 22 2022
%e A276550 Table starts:
%e A276550   1  2   3    4     5      6      7       8 ...
%e A276550   0  1   3    6    10     15     21      28 ...
%e A276550   0  2   7   16    30     50     77     112 ...
%e A276550   0  3  15   45   105    210    378     630 ...
%e A276550   0  6  36  132   372    882   1848    3528 ...
%e A276550   0  8  79  404  1460   4220  10423   22904 ...
%e A276550   0 16 195 1296  5890  20640  60021  151840 ...
%e A276550   0 24 477 4380 25275 107100 364854 1057392 ...
%e A276550   ...
%p A276550 A276550 := proc(n,k)
%p A276550     local d ;
%p A276550     add( numtheory[mobius](n/d)*A081720(d,k),d=numtheory[divisors](n)) ;
%p A276550 end proc:
%p A276550 seq(seq(A276550(n,d-n),n=1..d-1),d=2..10) ; # _R. J. Mathar_, Jan 22 2022
%t A276550 t[n_, k_] := Sum[EulerPhi[d] k^(n/d), {d, Divisors[n]}]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4;
%t A276550 T[n_, k_] := Sum[MoebiusMu[d] t[n/d, k], {d, Divisors[n]}];
%t A276550 Table[T[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Mar 26 2020 *)
%Y A276550 Columns 2-6 are A001371, A032294, A032295, A032296, A056347.
%Y A276550 Cf. A081720, A273891, A276543, A152176.
%K A276550 nonn,tabl,easy
%O A276550 1,2
%A A276550 _Andrew Howroyd_, Apr 09 2017