A241926 Table read by antidiagonals: T(n,k) (n >= 1, k >= 1) is the number of necklaces with n black beads and k white beads.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 4, 3, 1, 1, 3, 5, 5, 3, 1, 1, 4, 7, 10, 7, 4, 1, 1, 4, 10, 14, 14, 10, 4, 1, 1, 5, 12, 22, 26, 22, 12, 5, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 6, 19, 43, 66, 80, 66, 43, 19, 6, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1

Offset

1,5

Comments

Turning over the necklace is not allowed (the group is cyclic not dihedral). T(n,k) = T(k,n) follows immediately from the formula. - N. J. A. Sloane, May 03 2014

T(n, k) is the number of equivalence classes of k-tuples of residues modulo n, identifying those that differ componentwise by a constant and those that differ by a permutation. - Álvar Ibeas, Sep 21 2021

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.

A. Elashvili and M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9 (1998), no. 2, 233--238. MR1691428 (2000c:13006)

A. Elashvili, M. Jibladze and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173--188. MR1719140 (2000j:05009). See p. 174. - N. J. A. Sloane, Aug 06 2014

N. J. A. Sloane, A Note on Modular Partitions and Necklaces

Formula

T(n,k) = Sum_{d | gcd(n,k)} phi(d)*binomial((n+k)/d, n/d))/(n+k). [Corrected by N. J. A. Sloane, May 03 2014]

Example

The table starts:
  1, 1,  1,  1,  1,  1,   1,   1,   1,   1,   1,    1, ...
  1, 2,  2,  3,  3,  4,   4,   5,   5,   6,   6,    7, ...
  1, 2,  4,  5,  7, 10,  12,  15,  19,  22,  26,   31, ...
  1, 3,  5, 10, 14, 22,  30,  43,  55,  73,  91,  116, ...
  1, 3,  7, 14, 26, 42,  66,  99, 143, 201, 273,  364, ...
  1, 4, 10, 22, 42, 80, 132, 217, 335, 504, 728, 1038, ...
  ...

Maple

# Maple program for the table - N. J. A. Sloane, May 03 2014:
with(numtheory);
T:=proc(n, k) local d, s, g, t0;
t0:=0; s:=n+k; g:=gcd(n, k);
for d from 1 to s do
   if (g mod d) = 0 then t0:=t0+phi(d)*binomial(s/d, k/d); fi;
od: t0/s; end;
r:=n->[seq(T(n, k), k=1..12)];
[seq(r(n), n=1..12)];

Mathematica

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#] Binomial[(n+k)/#, n/#]& ]/ (n+k); Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 02 2015 *)

Prog

(PARI) T(n, k) = sumdiv(gcd(n, k), d, eulerphi(d)*binomial((n+k)\d, n\d))/(n+k)

Crossrefs

Same as A047996 with first row and main diagonal removed.

A037306 is yet another version.

Cf. A003239 (main diagonal).

See A245558, A245559 for a closely related array.

Sequence in context: A275298 A048570 A090806 * A174446 A071201 A318045

Adjacent sequences: A241923 A241924 A241925 * A241927 A241928 A241929

Keyword

nonn,tabl

Author

Franklin T. Adams-Watters, May 02 2014

Extensions

Edited by N. J. A. Sloane, May 03 2014

Elashvili et al. references supplied by Vladimir Popov, May 17 2014

Status

approved