Search: id:A047996
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%I A047996
%S A047996 1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,3,4,3,1,1,1,1,3,5,5,3,1,
               1,1,1,4,7,
%T A047996 10,7,4,1,1,1,1,4,10,14,14,10,4,1,1,1,1,5,12,22,26,22,12,5,1,1,1,1,5,
%U A047996 15,30,42,42,30,15,5,1,1,1,1,6,19,43,66,80,66,43,19,6,1,1,1,1,6,22
%N A047996 Triangle of circular binomial coefficients T(n,k), 0<=k<=n.
%C A047996 T(n,k)=number of necklaces with k black beads, n-k white beads.
%D A047996 D. E. Knuth, Computer science and its relation to mathematics, Amer. 
               Math. Monthly, 81 (1974), 323-343.
%D A047996 F. Ruskey and J. Sawada, "An Efficient Algorithm for Generating Necklaces 
               with Fixed Density", SIAM J. Computing, 29 (1999) 671-684.
%D A047996 H. S. Wilf, personal communication, Nov., 1990.
%D A047996 See A000031 for many additional references and links.
%H A047996 T. D. Noe, <a href="b047996.txt">Rows n=0..50 of triangle, flattened</
               a>
%H A047996 F. Ruskey, <a href="http://www.theory.cs.uvic.ca/~cos/gen/neck.html">
               Necklaces</a>
%H A047996 Petr Lisonek, <a href="http://www.lacim.uqam.ca/~plouffe/articles/thesis.pdf">
               Computer-assisted Studiesin Algebraic Combinatorics, pp. 72-73</a>
               .
%H A047996 Wikipedia, <a href="http://commons.wikimedia.org/wiki/Image:A31.gif">
               Necklaces Animation</a>.
%H A047996 Wolfram Research, <a href="http://demonstrations.wolfram.com/MakingNecklaces/
               ">Necklaces Applet</a>.
%H A047996 <a href="Sindx_Ne.html#necklaces">Index entries for sequences related 
               to necklaces</a>
%F A047996 T(n, k)=(1/n) * Sum_{d | (n, k)} phi(d)*binomial(n/d, k/d).
%F A047996 T(2*n,n)=A003239(n) . T(2*n+1,n)=A000108(n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Jul 25 2006
%e A047996 1; 1,1; 1,1,1; 1,1,1,1; 1,1,2,1,1; 1,1,2,2,1,1; 1,1,3,4,3,1,1; ...
%Y A047996 Row sums: A000031. Columns 0-12: A000012, A000012, A004526, A007997(n-3), 
               A008610, A008646, A032191-A032197.
%Y A047996 Cf. A051168, A052307, A052311-A052313.
%Y A047996 Sequence in context: A052307 A067059 A049704 this_sequence A063686 A008327 
               A133687
%Y A047996 Adjacent sequences: A047993 A047994 A047995 this_sequence A047997 A047998 
               A047999
%K A047996 nonn,tabl,easy,nice
%O A047996 0,13
%A A047996 N. J. A. Sloane (njas(AT)research.att.com).

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