A254593 -id:A254593 - cp's OEIS frontend

A268592 a(n) = (6/n^3) * Sum_{d|n} moebius(n/d)binomial(2d,d).

12, 3, 4, 6, 12, 25, 60, 150, 400, 1107, 3180, 9386, 28404, 87711, 275764, 880470, 2849916, 9336508, 30918732, 103384758, 348725540, 1185630123, 4060210764, 13996354586, 48541672872, 169293988125, 593488622344, 2090567755278, 7396924802052, 26281018091013, 93738717046476, 335563502259798

Offset: 1

Max Alekseyev, Feb 07 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1500
R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0 arXiv:1602.02632

Cf. A254593, A268618, A268619.

a[n_] := (6/n^3)* DivisorSum[n, MoebiusMu[n/#] Binomial[2 #, #] &]; Array[a, 50] (* G. C. Greubel, Dec 15 2017 *)

{ a(n) = sumdiv(n, d, moebius(n/d)*binomial(2*d, d))*6/n^3; }

a(n) = A007727(n)*6/n^3 = A045630(n)*12/n^3 = A060165(n)*6/n^2 = A022553(n)*12/n^2 = A268619(n)*6/n.

For n == 0, 1, or 3 (mod 4), a(n) = 2*A254593(n); for n == 2 (mod 4), a(n) = 2*A254593(n) - A254593(n/2)/2.

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

Original entry on oeis.org

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

Vladeta Jovovic, Oct 04 2007

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

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.

Cf. A022553, A145855, A254593.

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

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 *)

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

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

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

A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^ic(n,i)binomial(i*p,p) is divisible by p^(2n-1).

Original entry on oeis.org

1, 2, 1, 12, 9, 2, 60, 54, 20, 3, 840, 840, 400, 105, 12, 2520, 2700, 1500, 525, 108, 10, 27720, 31185, 19250, 8085, 2268, 385, 30, 360360, 420420, 280280, 133770, 45864, 10780, 1560, 105, 720720, 864864, 611520, 321048, 127008, 36960, 7488, 945, 56, 12252240, 15036840, 11138400, 6297480, 2776032, 942480

Offset: 1

Max Alekseyev, Feb 06 2016

			n=1: 1
n=2: 2, 1
n=3: 12, 9, 2
n=4: 60, 54, 20, 3
n=5: 840, 840, 400, 105, 12
...
For all primes p>3, p^3 divides 2 - binomial(2*p,p) (cf. A087754).
For all primes p>5, p^5 divides 12 - 9*binomial(2*p,p) + 2*binomial(3*p,p) (cf. A268589).
For all primes p>7, p^7 divides 60 - 54*binomial(2*p,p) + 20*binomial(3*p,p) - 3*binomial(4*p,p) (cf. A268590).

R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0; also arXiv, arXiv:1602.02632 [math.NT], 2016-2018.

Cf. A099996 (first column), A068550 (diagonal), A087754, A268589, A268590, A254593.

a3418[n_] := LCM @@ Range[n];
c[1, 1] = 1; c[n_, i_] := a3418[2(n-1)] Binomial[2n-1, n-i] ((2i-1)/i/ Binomial[2n-1, n]);
Table[c[n, i], {n, 1, 10}, {i, 1, n}] // Flatten (* Jean-François Alcover, Dec 04 2018 *)

{ A268512(n,i) = lcm(vector(2*(n-1),i,i)) * binomial(2*n-1,n-i) * (2*i-1) / i / binomial(2*n-1,n) }

c(n,i) = A003418(2*(n-1))*binomial(2*n-1,n-i)*(2*i-1)/i/binomial(2*n-1,n).

A268618 a(n) = (2/n^3) * Sum_{d|n} moebius(n/d)binomial(3d,d).

Original entry on oeis.org

6, 3, 6, 15, 48, 171, 678, 2871, 12858, 60084, 290814, 1448679, 7394106, 38527779, 204365880, 1101000087, 6013054788, 33239486925, 185736687366, 1047961118940, 5964676687668, 34219227608607, 197737647050742, 1150211467134927, 6731334034067058, 39614408616493581, 234342269725331130, 1392933275876114127

Offset: 1

Max Alekseyev, Feb 09 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1200
R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0 arXiv:1602.02632

Cf. A268592, A254593

a[n_] := (2/n^3)* DivisorSum[n, MoebiusMu[n/#] Binomial[3 #, #] &]; Array[a, 50] (* G. C. Greubel, Dec 15 2017 *)

{ a(n) = sumdiv(n, d, moebius(n/d)*binomial(3*d, d))*2/n^3; }

a(n) = (2/n^3)* Sum_{d|n} A008683(n/d)*A005809(d) = (2/n^2)*A060170(n) = (2/n)*A268617(n).

cp's OEIS Frontend

A268592 a(n) = (6/n^3) * Sum_{d|n} moebius(n/d)*binomial(2*d,d).

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

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A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^i*c(n,i)*binomial(i*p,p) is divisible by p^(2n-1).

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A268618 a(n) = (2/n^3) * Sum_{d|n} moebius(n/d)*binomial(3*d,d).

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Mathematica

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Formula

A268592 a(n) = (6/n^3) * Sum_{d|n} moebius(n/d)binomial(2d,d).

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

A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^ic(n,i)binomial(i*p,p) is divisible by p^(2n-1).

A268618 a(n) = (2/n^3) * Sum_{d|n} moebius(n/d)binomial(3d,d).