A050183 T(2n+5,n), array T as in A051168; a count of Lyndon words.

0, 1, 4, 15, 55, 200, 728, 2652, 9690, 35530, 130750, 482885, 1789515, 6653325, 24812400, 92798375, 347993910, 1308233790, 4929576600, 18615637950, 70441574000, 267058714626, 1014283603024, 3858687620200, 14702930414900

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• Index entries for sequences related to Lyndon words

Crossrefs

A diagonal of the square array described in A051168.

Programs

Maple

A050183 := proc(n)
    binomial(2*n+5,n) ;
    if modp(n,5) = 0 then
        %-binomial(2*n/5+1,n/5) ;
    end if;
    %/(2*n+5) ;
end proc:
seq(A050183(n),n=0..40) ; # R. J. Mathar, Oct 28 2021

PARI

a(n) = (1/(2*n+5))*sumdiv(gcd(n,5), d, moebius(d)*binomial((2*n+5)/d, n/d)); \\ Michel Marcus, Dec 05 2017

Formula

From Petros Hadjicostas, Dec 03 2017: (Start)

a(n) = (1/(2*n+5))*Sum_{d|gcd(n,5)} mu(d)*binomial((2*n+5)/d, n/d). (This is a special case of A. Howroyd's formula for double array A051168.)

a(n) = (1/(2*n+5))*(binomial(2*n+5, n) - binomial((2*n/5)+1, n/5)) if 5|n; = (1/(2*n+5))*binomial(2*n+5, n) otherwise.

(End)