login
Array read by antidiagonals: T(n,k) = number of primitive (period n) bracelets using a maximum of k different colored beads.
7

%I #30 Mar 30 2023 06:14:48

%S 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 50,105,132,79,16,0,9,28,77,210,372,404,195,24,0,10,36,112,378,882,

%U 1460,1296,477,42,0,11,45,156,630,1848,4220,5890,4380,1209,69,0

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

%C Turning over will not create a new bracelet.

%H Andrew Howroyd, <a href="/A276550/b276550.txt">Table of n, a(n) for n = 1..1275</a>

%H 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 F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>

%H F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]

%F T(n, k) = Sum_{d|n} mu(n/d) * A081720(d,k) for k<=n. Corrected Jan 22 2022

%e Table starts:

%e 1 2 3 4 5 6 7 8 ...

%e 0 1 3 6 10 15 21 28 ...

%e 0 2 7 16 30 50 77 112 ...

%e 0 3 15 45 105 210 378 630 ...

%e 0 6 36 132 372 882 1848 3528 ...

%e 0 8 79 404 1460 4220 10423 22904 ...

%e 0 16 195 1296 5890 20640 60021 151840 ...

%e 0 24 477 4380 25275 107100 364854 1057392 ...

%e ...

%p A276550 := proc(n,k)

%p local d ;

%p add( numtheory[mobius](n/d)*A081720(d,k),d=numtheory[divisors](n)) ;

%p end proc:

%p seq(seq(A276550(n,d-n),n=1..d-1),d=2..10) ; # _R. J. Mathar_, Jan 22 2022

%t 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 T[n_, k_] := Sum[MoebiusMu[d] t[n/d, k], {d, Divisors[n]}];

%t Table[T[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Mar 26 2020 *)

%Y Columns 2-6 are A001371, A032294, A032295, A032296, A056347.

%Y Cf. A081720, A273891, A276543, A152176.

%K nonn,tabl,easy

%O 1,2

%A _Andrew Howroyd_, Apr 09 2017