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Search: id:A047996
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| A047996 |
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Triangle of circular binomial coefficients T(n,k), 0<=k<=n. |
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+0
19
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| 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, 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, 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
(list; table; graph; listen)
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OFFSET
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0,13
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COMMENT
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T(n,k)=number of necklaces with k black beads, n-k white beads.
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REFERENCES
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D. E. Knuth, Computer science and its relation to mathematics, Amer. Math. Monthly, 81 (1974), 323-343.
F. Ruskey and J. Sawada, "An Efficient Algorithm for Generating Necklaces with Fixed Density", SIAM J. Computing, 29 (1999) 671-684.
H. S. Wilf, personal communication, Nov., 1990.
See A000031 for many additional references and links.
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LINKS
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T. D. Noe, Rows n=0..50 of triangle, flattened
F. Ruskey, Necklaces
Petr Lisonek, Computer-assisted Studiesin Algebraic Combinatorics, pp. 72-73.
Wikipedia, Necklaces Animation.
Wolfram Research, Necklaces Applet.
Index entries for sequences related to necklaces
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FORMULA
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T(n, k)=(1/n) * Sum_{d | (n, k)} phi(d)*binomial(n/d, k/d).
T(2*n,n)=A003239(n) . T(2*n+1,n)=A000108(n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jul 25 2006
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EXAMPLE
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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; ...
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CROSSREFS
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Row sums: A000031. Columns 0-12: A000012, A000012, A004526, A007997(n-3), A008610, A008646, A032191-A032197.
Cf. A051168, A052307, A052311-A052313.
Sequence in context: A052307 A067059 A049704 this_sequence A063686 A008327 A133687
Adjacent sequences: A047993 A047994 A047995 this_sequence A047997 A047998 A047999
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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