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A053727
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Triangle T(n,k) = Sum_{d|gcd(n,k)} mu(d)*C(n/d,k/d) (n >= 1, 1 <= k <= n).
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2
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1, 2, 0, 3, 3, 0, 4, 4, 4, 0, 5, 10, 10, 5, 0, 6, 12, 18, 12, 6, 0, 7, 21, 35, 35, 21, 7, 0, 8, 24, 56, 64, 56, 24, 8, 0, 9, 36, 81, 126, 126, 81, 36, 9, 0, 10, 40, 120, 200, 250, 200, 120, 40, 10, 0, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 0, 12, 60
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OFFSET
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1,2
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COMMENTS
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Triangle of number of primitive words over {0,1} of length n that contain k 1's, for n,k >= 1. - Benoit Cloitre, Jun 08 2004
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic sequences, Cambridge University Press, 2003, p. 29.
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LINKS
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EXAMPLE
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Triangle begins
1;
2, 0;
3, 3, 0;
4, 4, 4, 0;
5, 10, 10, 5, 0;
6, 12, 18, 12, 6, 0;
...
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MATHEMATICA
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T[n_, k_] := DivisorSum[GCD[k, n], MoebiusMu[#] Binomial[n/#, k/#] &]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 02 2015 *)
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PROG
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(PARI) T(n, k)=sumdiv(gcd(k, n), d, moebius(d)*binomial(n/d, k/d)) \\ Benoit Cloitre, Jun 08 2004
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CROSSREFS
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Same triangle as A050186 except this one does not include column 0.
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KEYWORD
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AUTHOR
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STATUS
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approved
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