A131868 a(n) = (2*n^2)^(-1)*Sum_{d|n} (-1)^(n+d)*moebius(n/d)*binomial(2*d,d).

1, 1, 1, 2, 5, 13, 35, 100, 300, 925, 2915, 9386, 30771, 102347, 344705, 1173960, 4037381, 14004912, 48954659, 172307930, 610269695, 2173656683, 7782070631, 27992709172, 101128485150, 366803656323, 1335349400274, 4877991428982

Offset: 1

Author

Vladeta Jovovic, Oct 04 2007

Keywords

• easy
• nonn

Comments

n*a(n) is the number of n-member subsets of {1,2,3,...,2*n-1} that sum to 1 mod n, cf. A145855. - Vladeta Jovovic, Oct 28 2008

a(n) is the number of orbits under the S_n action on a set closely related to the set of parking functions. See Konvalinka-Tewari reference below. - Vasu Tewari, Mar 17 2020

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Kunal Gupta and Pietro Longhi, Vortices on Cylinders and Warped Exponential Networks, arXiv:2407.08445 [hep-th], 2024. See pp. 41, 49.
• M. Kontsevich, R. Stanley, O. Gorodetsky, et al. A congruence involving binomial coefficients, Mathoverflow, 2015.
• Matjaž Konvalinka and Vasu Tewari, Some natural extensions of the parking space, arXiv:2003.04134 [math.CO], 2020.
• Jerome Malenfant, On the Matrix-Element Expansion of a Circulant Determinant, arXiv:1502.06012 [math.NT], 2015.
• Steven Rayan, Aspects of the topology and combinatorics of Higgs bundle moduli spaces, arXiv:1809.05732 [math.AG], 2018.

Crossrefs

Cf. A022553, A145855, A254593.

Programs

Maple

A131868 := proc(n) local a,d ; a := 0 ; for d in numtheory[divisors](n) do a := a+(-1)^(n+d)*numtheory[mobius](n/d)*binomial(2*d,d) ; od: a/2/n^2 ; end: seq(A131868(n),n=1..30) ; # R. J. Mathar, Oct 24 2007

Mathematica

a = {}; For[n = 1, n < 30, n++, b = Divisors[n]; s = 0; For[j = 1, j < Length[b] + 1, j++, s = s + (-1)^(n + b[[j]])*MoebiusMu[n/b[[j]]]* Binomial[2*b[[j]], b[[j]]]]; AppendTo[a, s/(2*n^2)]]; a (* Stefan Steinerberger, Oct 26 2007 *)
a[n_] := 1/(2n^2) DivisorSum[n, (-1)^(n+#) MoebiusMu[n/#] Binomial[2#, #]& ]; Array[a, 30] (* Jean-François Alcover, Dec 18 2015 *)

PARI

a(n) = (2*n^2)^(-1)*sumdiv(n, d, (-1)^(n+d)*moebius(n/d)*binomial(2*d,d)); \\ Michel Marcus, Dec 06 2018

Formula

a(n) ~ 2^(2*n - 1) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Jun 08 2019

Extensions

More terms from R. J. Mathar and Stefan Steinerberger, Oct 24 2007