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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
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
KEYWORD
nonn,tabl
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, May 03 2014
Elashvili et al. references supplied by Vladimir Popov, May 17 2014
STATUS
approved

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Last modified April 20 06:53 EDT 2024. Contains 371799 sequences. (Running on oeis4.)