%I #46 Jul 16 2024 15:56:11
%S 1,1,1,2,5,13,35,100,300,925,2915,9386,30771,102347,344705,1173960,
%T 4037381,14004912,48954659,172307930,610269695,2173656683,7782070631,
%U 27992709172,101128485150,366803656323,1335349400274,4877991428982
%N a(n) = (2*n^2)^(-1)*Sum_{d|n} (-1)^(n+d)*moebius(n/d)*binomial(2*d,d).
%C 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
%C 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
%H G. C. Greubel, <a href="/A131868/b131868.txt">Table of n, a(n) for n = 1..1000</a>
%H Kunal Gupta and Pietro Longhi, <a href="https://arxiv.org/abs/2407.08445">Vortices on Cylinders and Warped Exponential Networks</a>, arXiv:2407.08445 [hep-th], 2024. See pp. 41, 49.
%H M. Kontsevich, R. Stanley, O. Gorodetsky, et al. <a href="http://mathoverflow.net/q/195339">A congruence involving binomial coefficients</a>, Mathoverflow, 2015.
%H Matjaž Konvalinka and Vasu Tewari, <a href="https://arxiv.org/abs/2003.04134">Some natural extensions of the parking space</a>, arXiv:2003.04134 [math.CO], 2020.
%H Jerome Malenfant, <a href="http://arxiv.org/abs/1502.06012">On the Matrix-Element Expansion of a Circulant Determinant</a>, arXiv:1502.06012 [math.NT], 2015.
%H Steven Rayan, <a href="https://arxiv.org/abs/1809.05732">Aspects of the topology and combinatorics of Higgs bundle moduli spaces</a>, arXiv:1809.05732 [math.AG], 2018.
%F a(n) ~ 2^(2*n - 1) / (sqrt(Pi) * n^(5/2)). - _Vaclav Kotesovec_, Jun 08 2019
%p 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
%t 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 *)
%t a[n_] := 1/(2n^2) DivisorSum[n, (-1)^(n+#) MoebiusMu[n/#] Binomial[2#, #]& ]; Array[a, 30] (* _Jean-François Alcover_, Dec 18 2015 *)
%o (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
%Y Cf. A022553, A145855, A254593.
%K easy,nonn
%O 1,4
%A _Vladeta Jovovic_, Oct 04 2007
%E More terms from _R. J. Mathar_ and _Stefan Steinerberger_, Oct 24 2007