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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.

%I A241926 #51 Nov 09 2024 02:36:29
%S A241926 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 A241926 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 A241926 66,80,66,43,19,6,1,1,6,22,55,99,132,132,99,55,22,6,1
%N 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.
%C A241926 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
%C A241926 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
%H A241926 G. C. Greubel, <a href="/A241926/b241926.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A241926 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 A241926 A. Elashvili and 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 A241926 A. Elashvili, M. Jibladze and 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 A241926 N. J. A. Sloane, <a href="/A047996/a047996.pdf">A Note on Modular Partitions and Necklaces</a>
%F A241926 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 A241926 The table starts:
%e A241926   1, 1,  1,  1,  1,  1,   1,   1,   1,   1,   1,    1, ...
%e A241926   1, 2,  2,  3,  3,  4,   4,   5,   5,   6,   6,    7, ...
%e A241926   1, 2,  4,  5,  7, 10,  12,  15,  19,  22,  26,   31, ...
%e A241926   1, 3,  5, 10, 14, 22,  30,  43,  55,  73,  91,  116, ...
%e A241926   1, 3,  7, 14, 26, 42,  66,  99, 143, 201, 273,  364, ...
%e A241926   1, 4, 10, 22, 42, 80, 132, 217, 335, 504, 728, 1038, ...
%e A241926   ...
%p A241926 # Maple program for the table - _N. J. A. Sloane_, May 03 2014:
%p A241926 with(numtheory);
%p A241926 T:=proc(n,k) local d, s, g, t0;
%p A241926 t0:=0; s:=n+k; g:=gcd(n,k);
%p A241926 for d from 1 to s do
%p A241926    if (g mod d) = 0 then t0:=t0+phi(d)*binomial(s/d,k/d); fi;
%p A241926 od: t0/s; end;
%p A241926 r:=n->[seq(T(n,k),k=1..12)];
%p A241926 [seq(r(n),n=1..12)];
%t A241926 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 *)
%o A241926 (PARI) T(n,k) = sumdiv(gcd(n,k),d,eulerphi(d)*binomial((n+k)\d,n\d))/(n+k)
%Y A241926 Same as A047996 with first row and main diagonal removed.
%Y A241926 A037306 is yet another version.
%Y A241926 Cf. A003239 (main diagonal).
%Y A241926 See A245558, A245559 for a closely related array.
%K A241926 nonn,tabl
%O A241926 1,5
%A A241926 _Franklin T. Adams-Watters_, May 02 2014
%E A241926 Edited by _N. J. A. Sloane_, May 03 2014
%E A241926 Elashvili et al. references supplied by Vladimir Popov, May 17 2014