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Search: id:A121775
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| A121775 |
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Triangle, read by rows, defined by: T(n,k) = Sum_{d|n} phi(n/d)*C(d,k) for n>0, k>=0, with T(0,0) = 1. |
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+0
2
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| 1, 1, 1, 2, 3, 1, 3, 5, 3, 1, 4, 8, 7, 4, 1, 5, 9, 10, 10, 5, 1, 6, 15, 20, 21, 15, 6, 1, 7, 13, 21, 35, 35, 21, 7, 1, 8, 20, 36, 60, 71, 56, 28, 8, 1, 9, 21, 42, 86, 126, 126, 84, 36, 9, 1, 10, 27, 59, 130, 215, 253, 210, 120, 45, 10, 1, 11, 21, 55, 165, 330, 462, 462, 330, 165, 55
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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For n>0, (1/n)*Sum_{k=0..n} T(n,k)*(c-1)^k = number of n-bead necklaces with c colors: A000031 (c=2), A001867 (c=3), A001868 (c=4), A001869 (c=5), A054625 (c=6), A054626 (c=7), A054627 (c=8), A054628 (c=9), A054629 (c=10).
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EXAMPLE
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Triangle begins:
1;
1, 1;
2, 3, 1;
3, 5, 3, 1;
4, 8, 7, 4, 1;
5, 9, 10, 10, 5, 1;
6, 15, 20, 21, 15, 6, 1;
7, 13, 21, 35, 35, 21, 7, 1;
8, 20, 36, 60, 71, 56, 28, 8, 1;
9, 21, 42, 86, 126, 126, 84, 36, 9, 1;
10, 27, 59, 130, 215, 253, 210, 120, 45, 10, 1;
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PROGRAM
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(PARI) T(n, k)=if(n<k|k<0, 0, if(n==0, 1, sumdiv(n, d, eulerphi(n/d)*binomial(d, k))))
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CROSSREFS
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Cf. A053635 (row sums), A121776 (antidiagonal sums), A054630.
Sequence in context: A154722 A035517 A099471 this_sequence A127951 A162609 A054250
Adjacent sequences: A121772 A121773 A121774 this_sequence A121776 A121777 A121778
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Aug 23 2006
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