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A106519
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a(n) = (2/n)*binomial(2*n-2, n-1) - (1/(2*n))*Sum_{d|n} Moebius(d)*binomial(2*n/d, n/d).
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1
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1, 1, 1, 2, 3, 9, 19, 58, 160, 499, 1527, 4940, 16001, 53187, 178305, 606330, 2079863, 7203864, 25138879, 88367780, 312577245, 1112119079, 3977502767, 14294207172, 51596165898, 186998138529, 680272336906, 2483341820512, 9094756956909
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = (2/n)*binomial(2*n-2, n-1) - (1/(2*n))*Sum_{d|n} Moebius(d)*binomial(2*n/d, n/d).
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MAPLE
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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)) );
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MATHEMATICA
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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 *)
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PROG
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(Sage)
def a(n):
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
(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
(Magma)
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)]) >;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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