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

%I A050183 #19 Oct 28 2021 10:56:40
%S A050183 0,1,4,15,55,200,728,2652,9690,35530,130750,482885,1789515,6653325,
%T A050183 24812400,92798375,347993910,1308233790,4929576600,18615637950,
%U A050183 70441574000,267058714626,1014283603024,3858687620200,14702930414900
%N A050183 T(2n+5,n), array T as in A051168; a count of Lyndon words.
%H A050183 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%F A050183 From _Petros Hadjicostas_, Dec 03 2017: (Start)
%F A050183 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.)
%F A050183 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.
%F A050183 (End)
%p A050183 A050183 := proc(n)
%p A050183     binomial(2*n+5,n) ;
%p A050183     if modp(n,5) = 0 then
%p A050183         %-binomial(2*n/5+1,n/5) ;
%p A050183     end if;
%p A050183     %/(2*n+5) ;
%p A050183 end proc:
%p A050183 seq(A050183(n),n=0..40) ; # _R. J. Mathar_, Oct 28 2021
%o A050183 (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
%Y A050183 A diagonal of the square array described in A051168.
%K A050183 nonn
%O A050183 0,3
%A A050183 _Clark Kimberling_