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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. 8

%I

%S 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,

%T 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,

%U 66,80,66,43,19,6,1,1,6,22,55,99,132,132,99,55,22,6,1

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

%C 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

%H G. C. Greubel, <a href="/A241926/b241926.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Paul Drube and Puttipong Pongtanapaisan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Drube/drube3.html">Annular Non-Crossing Matchings</a>, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.

%H A. Elashvili, M. Jibladze, <a href="http://dx.doi.org/10.1016/S0019-3577(98)80021-9">Hermite reciprocity for the regular representations of cyclic groups</a>, Indag. Math. (N.S.) 9 (1998), no. 2, 233--238. MR1691428 (2000c:13006)

%H A. Elashvili, M. Jibladze, D. Pataraia, <a href="http://dx.doi.org/10.1023/A:1018727630642">Combinatorics of necklaces and "Hermite reciprocity"</a>, J. Algebraic Combin. 10 (1999), no. 2, 173--188. MR1719140 (2000j:05009). See p. 174. - _N. J. A. Sloane_, Aug 06 2014

%H N. J. A. Sloane, <a href="/A047996/a047996.pdf">A Note on Modular Partitions and Necklaces</a>

%F 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]

%e The table starts:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

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

%e 1, 2, 4, 5, 7, 10, 12, 15, 19, 22, 26, 31, ...

%e 1, 3, 5, 10, 14, 22, 30, 43, 55, 73, 91, 116, ...

%e 1, 3, 7, 14, 26, 42, 66, 99, 143, 201, 273, 364, ...

%e 1, 4, 10, 22, 42, 80, 132, 217, 335, 504, 728, 1038, ...

%e ...

%p # Maple program for the table - _N. J. A. Sloane_, May 03 2014:

%p with(numtheory);

%p T:=proc(n,k) local d, s, g, t0;

%p t0:=0; s:=n+k; g:=gcd(n,k);

%p for d from 1 to s do

%p if (g mod d) = 0 then t0:=t0+phi(d)*binomial(s/d,k/d); fi;

%p od: t0/s; end;

%p r:=n->[seq(T(n,k),k=1..12)];

%p [seq(r(n),n=1..12)];

%t 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 *)

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

%Y Same as A047996 with first row and main diagonal removed.

%Y A037306 is yet another version.

%Y Cf. A003239 (main diagonal).

%Y See A245558, A245559 for a closely related array.

%K nonn,tabl

%O 1,5

%A _Franklin T. Adams-Watters_, May 02 2014

%E Edited by _N. J. A. Sloane_, May 03 2014

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

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Last modified October 17 01:37 EDT 2018. Contains 316275 sequences. (Running on oeis4.)