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).

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

Offset: 1

Author

F. Chapoton, May 30 2005

Keywords

• nonn
• easy
• changed

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000108, A001761, A008683, A098796.

Programs

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)]) >;
[A106519(n): n in [1..30]]; // G. C. Greubel, Aug 06 2021

Maple

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)) );
seq(a(n), n = 1..30); # modified by G. C. Greubel, Aug 06 2021

Mathematica

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 *)

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

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

Formula

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

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

a(prime(n)) = A098796(n). - Peter Bala, Aug 20 2025

Extensions

More terms from Robert G. Wilson v, May 31 2005

a(1) = 1 prepended by G. C. Greubel, Aug 06 2021