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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. 9
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

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, 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, 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-Fran├žois 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

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Last modified May 14 13:50 EDT 2021. Contains 343884 sequences. (Running on oeis4.)