A050181 T(2n+3, n), array T as in A051168; a count of Lyndon words.

0, 1, 3, 9, 30, 99, 333, 1144, 3978, 13995, 49742, 178296, 643842, 2340135, 8554275, 31429026, 115997970, 429874830, 1598952366, 5967382200, 22338765540, 83859016527, 315614844558, 1190680751376, 4501802223090, 17055399281284

Offset

0,3

Links

Table of n, a(n) for n=0..25.

Index entries for sequences related to Lyndon words

Formula

Conjecture: -(n-1)*(n+3)*(n+2)*a(n) + 2*(3*n-4)*(n+2)*(n+1)*a(n-1) - 4*n*(n+1)*(2*n-5)*a(n-2) + 2*(n-1)*(n+2)*(2*n-3)*a(n-3) - 4*(2*n-5)*(3*n-4)*(n+1)*a(n-4) + 8*n*(2*n-5)*(2*n-7)*a(n-5) = 0. - R. J. Mathar, Jul 20 2016

From Petros Hadjicostas, Nov 16 2017: (Start)

a(n) = (1/(2*n+3))*Sum_{d|gcd(n,3)} mu(d)*binomial((2*n+3)/d, n/d). (This is a special case of A. Howroyd's formula for double array A051168.)

a(n) = (1/(2*n+3))*(binomial(2*n+3, n) - binomial((2*n/3)+1, n/3)) if 3|n; = (1/(2*n+3))*binomial(2*n+3, n) otherwise.

Using the above formulae, one can verify R. J. Mathar's conjecture above.

(End)

Maple

A050181 := proc(n)
    A051168(2*n+3, n) ;
end proc: # R. J. Mathar, Jul 20 2016

Mathematica

a[n_] := (1/(2n+3)) Sum[MoebiusMu[d] Binomial[(2n+3)/d, n/d], {d, Divisors[ GCD[n, 3]]}];
a /@ Range[0, 25] (* Jean-François Alcover, Sep 17 2019, from PARI *)

Prog

(PARI) a(n) = (1/(2*n+3))*sumdiv(gcd(n, 3), d, moebius(d)*binomial((2*n+3)/d, n/d)); \\ Michel Marcus, Nov 18 2017

Crossrefs

Cf. A003441.

A diagonal of the square array described in A051168.

Sequence in context: A089978 A052906 A102898 * A275690 A089931 A148946

Adjacent sequences: A050178 A050179 A050180 * A050182 A050183 A050184

Keyword

nonn

Author

Clark Kimberling

Status

approved