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A103306
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Triangle read by rows: T(n,k) = number of k-subsets of the n-th roots of 1 that add to zero (0 <= k <= n).
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2
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1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 3, 2, 3, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 0, 0, 3, 0, 0, 3, 0, 0, 1, 1, 0, 5, 0, 10, 2, 10, 0, 5, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 6, 4, 15, 12, 24, 12, 15, 4, 6, 0, 1, 1, 0, 0, 0, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,13
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COMMENTS
| Observe that T(n,k) = binomial(n,k) (mod n). Because the sum of the n n-th roots of unity is 0 for n>1, each row is symmetric for n>1. Hence only k=0..floor(n/2) need to be computed. - T. D. Noe (noe(AT)sspectra.com), Jan 16 2008
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LINKS
| Wouter Meeussen and T. D. Noe, Rows n=0..43 of triangle, flattened
Wouter Meeussen, More terms
Gary Sivek, On vanishing sums of distinct roots of unity, #A31, Integers 10 (2010), 365-368.
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EXAMPLE
| Triangle begins:
{1}
{1, 0},
{1, 0, 1},
{1, 0, 0, 1},
{1, 0, 2, 0, 1},
{1, 0, 0, 0, 0, 1},
{1, 0, 3, 2, 3, 0, 1},
{1, 0, 0, 0, 0, 0, 0, 1},
{1, 0, 4, 0, 6, 0, 4, 0, 1},
{1, 0, 0, 3, 0, 0, 3, 0, 0, 1},
T(10,4)=10, counting {1,2,6,7},{1,3,6,8},{1,4,6,9},{1,5,6,10},{2,3,7,8},{2,4,7,9},{2,5,7,10},{3,4,8,9},{3,5,8,10},{4,5,9,10}.
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MATHEMATICA
| <<DiscreteMath`Combinatorica`; Table[Count[(KSubsets[Range[n], k]), q_List/; Chop[Plus@@(E^(2.*Pi*I*q/n))]===0], {n, 0, 24}, {k, 0, n}]
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CROSSREFS
| Row sums give A103314. Cf. A070894.
Sequence in context: A204246 A053200 A050870 * A163510 A124735 A064874
Adjacent sequences: A103303 A103304 A103305 * A103307 A103308 A103309
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KEYWORD
| nonn,tabl
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AUTHOR
| Wouter Meeussen (wouter.meeussen(AT)pandora.be), Mar 11 2005
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