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A053727 Triangle T(n,k) = Sum_{d|gcd(n,k)} mu(d)*C(n/d,k/d) (n >= 1, 1 <= k <= n).

%I A053727 #26 Oct 20 2019 01:55:02
%S A053727 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,
%T A053727 24,56,64,56,24,8,0,9,36,81,126,126,81,36,9,0,10,40,120,200,250,200,
%U A053727 120,40,10,0,11,55,165,330,462,462,330,165,55,11,0,12,60
%N A053727 Triangle T(n,k) = Sum_{d|gcd(n,k)} mu(d)*C(n/d,k/d) (n >= 1, 1 <= k <= n).
%C A053727 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
%D A053727 J.-P. Allouche and J. Shallit, Automatic sequences, Cambridge University Press, 2003, p. 29.
%H A053727 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%e A053727 Triangle begins
%e A053727   1;
%e A053727   2,  0;
%e A053727   3,  3,  0;
%e A053727   4,  4,  4,  0;
%e A053727   5, 10, 10,  5,  0;
%e A053727   6, 12, 18, 12,  6,  0;
%e A053727   ...
%t A053727 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 *)
%o A053727 (PARI) T(n,k)=sumdiv(gcd(k,n),d,moebius(d)*binomial(n/d,k/d)) \\ _Benoit Cloitre_, Jun 08 2004
%Y A053727 Cf. A042979, A042980. T(2n, n), T(2n+1, n) match A007727, A001700, respectively. Row sums match A027375.
%Y A053727 Same triangle as A050186 except this one does not include column 0.
%K A053727 nonn,tabl
%O A053727 1,2
%A A053727 _N. J. A. Sloane_, Mar 24 2000