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

Author

Franklin T. Adams-Watters, May 02 2014

Keywords

• nonn
• tabl

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

Examples

			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, ...
  ...
		

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

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.

Programs

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 *)

PARI

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

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]

Extensions

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

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