A106520 a(n) = A068875(n-1) - A003239(n).

1, 0, 0, 0, 2, 4, 18, 48, 156, 472, 1526, 4852, 16000, 52940, 178276, 605520, 2079862, 7201084, 25138878, 88358520, 312576996, 1112087012, 3977502766, 14294093652, 51596165872, 186997738504, 680272334202, 2483340387644, 9094756956908

Offset: 1

Author

F. Chapoton, May 30 2005

Keywords

• nonn

Comments

This is the multiplicity of the trivial module in a 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)!) (Catalan) over the cyclic groups C_n.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• F. Chapoton, On some anticyclic operads, Algebraic and Geometric Topology 5 (2005), paper no. 4, pages 53-69.

Crossrefs

Cf. A000010, A000108, A001761.

Programs

Magma

A106520:= func< n | 2*Catalan(n-1) - (1/(2*n))*(&+[Round(Gamma(2*n/d +1)/Gamma(n/d +1)^2)*EulerPhi(d): d in Divisors(n)]) >;
[A106520(n): n in [1..40]]; // G. C. Greubel, Aug 06 2021

Maple

with(numtheory);
a:= proc(n) (2/n)*binomial(2*n-2, n-1) - (1/(2*n))*add(phi(d)*binomial(2*n/d, n/d), d = divisors(n)) end:
seq(a(n), n = 1..40);

Mathematica

a[n_]:= 2/n*Binomial[2*n-2, n-1] - 1/(2*n)*DivisorSum[n, EulerPhi[#]* Binomial[2*n/#, n/#]&]; Table[a[n], {n, 40}] (* Jean-François Alcover, Feb 20 2017 *)

PARI

a(n) = (2/n) * binomial(2*n-2, n-1) - 1/(2*n) * sumdiv(n, d, eulerphi(d) * binomial(2*n/d, n/d)); \\ Michel Marcus, Aug 08 2021

Sage

def a(n): return 2*catalan_number(n-1) - (1/(2*n))*sum(euler_phi(n/d)*binomial(2*d, d) for d in divisors(n))
[a(n) for n in (1..40)] # G. C. Greubel, Aug 06 2021

Formula

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

a(n) = 2*A000108(n-1) - (1/(2*n))*Sum_{d divides n} (n/d + 1)*A000108(n/d) * A000010(d). - G. C. Greubel, Aug 06 2021

Extensions

Terms a(1) to a(4) prepended by G. C. Greubel, Aug 06 2021