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

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

Offset: 1

Author

N. J. A. Sloane, Mar 24 2000

Keywords

• nonn
• tabl

Comments

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

Examples

			Triangle begins
  1;
  2,  0;
  3,  3,  0;
  4,  4,  4,  0;
  5, 10, 10,  5,  0;
  6, 12, 18, 12,  6,  0;
  ...
		

References

• J.-P. Allouche and J. Shallit, Automatic sequences, Cambridge University Press, 2003, p. 29.

Links

• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A042979, A042980. T(2n, n), T(2n+1, n) match A007727, A001700, respectively. Row sums match A027375.

Same triangle as A050186 except this one does not include column 0.

Programs

Mathematica

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 *)

PARI

T(n,k)=sumdiv(gcd(k,n),d,moebius(d)*binomial(n/d,k/d)) \\ Benoit Cloitre, Jun 08 2004