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A106519 a(n) = (2/n)*binomial(2*n-2, n-1) - (1/(2*n))*Sum_{d|n} Moebius(d)*binomial(2*n/d, n/d).

%I A106519 #27 Aug 27 2025 01:15:17
%S A106519 1,1,1,2,3,9,19,58,160,499,1527,4940,16001,53187,178305,606330,
%T A106519 2079863,7203864,25138879,88367780,312577245,1112119079,3977502767,
%U A106519 14294207172,51596165898,186998138529,680272336906,2483341820512,9094756956909
%N A106519 a(n) = (2/n)*binomial(2*n-2, n-1) - (1/(2*n))*Sum_{d|n} Moebius(d)*binomial(2*n/d, n/d).
%C A106519 A simple formula with no known combinatorial interpretation. This should give the multiplicity of the trivial module in some sequence of modules of dimension (2*n-2)!/n! over the symmetric groups S_n induced from modules of dimension (2*n-2)!/n!(n-1)! over the cyclic groups C_n.
%H A106519 G. C. Greubel, <a href="/A106519/b106519.txt">Table of n, a(n) for n = 1..1000</a>
%F A106519 a(n) = (2/n)*binomial(2*n-2, n-1) - (1/(2*n))*Sum_{d|n} Moebius(d)*binomial(2*n/d, n/d).
%F A106519 a(n) = 2*A000108(n-1) - (1/(2*n))*Sum_{d|n} Moebius(d)*(n/d + 1)*A000108(n/d). - _G. C. Greubel_, Aug 06 2021
%F A106519 a(prime(n)) = A098796(n). - _Peter Bala_, Aug 20 2025
%p A106519 a:= n -> (2/n)*( binomial(2*n-2, n-1) - (1/4)*add(NumberTheory[Moebius](d)*binomial(2*n/d, n/d), d = Divisors(n)) );
%p A106519 seq(a(n), n = 1..30); # modified by _G. C. Greubel_, Aug 06 2021
%t A106519 f[n_] := Block[{d = Divisors[n]}, 2*Binomial[2n-2, n-1]/n - Plus @@ (MoebiusMu[d]*Binomial[2*n/d, n/d])/(2n)]; Table[f[n], {n, 29}] (* _Robert G. Wilson v_, May 31 2005 *)
%o A106519 (Sage)
%o A106519 def a(n):
%o A106519     return binomial(2*n-2,n-1)*2//n - sum(moebius(n//d)*binomial(2*d,d) for d in divisors(n))//(2*n) # _F. Chapoton_, May 31 2020
%o A106519 (PARI) a(n) = (2*binomial(2*n-2, n-1) - sumdiv(n, d, moebius(d)*binomial(2*n/d, n/d))/2)/n; \\ _Michel Marcus_, Aug 06 2021
%o A106519 (Magma)
%o A106519 A106519:= func< n | 2*Catalan(n-1) - (1/(2*n))*(&+[Round(Gamma(2*n/d +1)/Gamma(n/d +1)^2)*MoebiusMu(d): d in Divisors(n)]) >;
%o A106519 [A106519(n): n in [1..30]]; // _G. C. Greubel_, Aug 06 2021
%Y A106519 Cf. A000108, A001761, A008683, A098796.
%K A106519 nonn,easy,changed
%O A106519 1,4
%A A106519 _F. Chapoton_, May 30 2005
%E A106519 More terms from _Robert G. Wilson v_, May 31 2005
%E A106519 a(1) = 1 prepended by _G. C. Greubel_, Aug 06 2021