A022553 -id:A022553 - cp's OEIS frontend

A131868 a(n) = (2n^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

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

A268619 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)binomial(2d,d).

Original entry on oeis.org

2, 1, 2, 4, 10, 25, 70, 200, 600, 1845, 5830, 18772, 61542, 204659, 689410, 2347920, 8074762, 28009524, 97909318, 344615860, 1220539390, 4347310451, 15564141262, 55985418344, 202256970300, 733607281875, 2670698800548, 9755982857964, 35751803209918, 131405090455065, 484316704740126, 1789672012052256

Offset: 1

Max Alekseyev, Feb 09 2016

nonn

6*a(n) is divisible by n (cf. A268592).

Cf. A000984, A007727, A008683, A022553, A045630, A060165, A268592, A268617.

a[n_] := DivisorSum[n, MoebiusMu[n/#] * Binomial[2*#, #] &] / n^2; Array[a, 35] (* Amiram Eldar, Aug 24 2023 *)

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

a(n) = (1/n^2)* Sum_{d|n} A008683(n/d)*A000984(d).

a(n) = A007727(n)/n^2 = A045630(n)*2/n^2 = A060165(n)/n = A022553(n)*2/n.

A178904 This should be related to the Coxeter transformations of the posets of partitions in rectangular boxes of size m times n.

Original entry on oeis.org

1, -1, -1, 0, -1, 0, 0, 1, 1, 0, 0, -1, 1, -1, 0, 0, 1, -1, -1, 1, 0, 0, -1, 2, -3, 2, -1, 0, 0, 1, -3, 4, 4, -3, 1, 0, 0, -1, 3, -6, 8, -6, 3, -1, 0, 0, 1, -3, 9, -13, -13, 9, -3, 1, 0, 0, -1, 4, -11, 19, -23, 19, -11, 4, -1, 0, 0, 1, -5, 13, -27, 39, 39, -27, 13, -5, 1, 0, 0, -1, 5, -17, 38, -61, 71, -61, 38, -17, 5, -1, 0

Offset: 0

F. Chapoton, Jun 22 2010

This table is symmetric: a(m,n)=a(n,m) for all m,n>=0.

			a(0,0) = 1, a(1,0) = a(0,1) = -1.
Triangle begins:
   1;
  -1, -1;
   0, -1,  0;
   0,  1,  1,  0;
   0, -1,  1, -1,  0;
   0,  1, -1, -1,  1,  0;
   0, -1,  2, -3,  2, -1, 0;
   ...

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A178738, A178749, A022553, A131868, A163210.

b[m_, n_] := (-1)^Max[m, n]*Binomial[m+n, n]; A[m_, n_] := DivisorSum[ n+m+1, b[Floor[m/#], Floor[n/#]]*MoebiusMu[#]&]/(m+n+1); Table[A[m-n, n], {m, 0, 12}, {n, 0, m}] // Flatten (* Jean-François Alcover, Feb 23 2017, adapted from Python *)

def twisted_binomial(m, n):
    return (-1)**max(m, n) * binomial(m + n, n)
def coefficients_A(m, n):
    return sum(twisted_binomial(m // d, n // d) * moebius(d)
           for d in divisors(m + n + 1)) / (m + n + 1)
matrix(ZZ, 8, 8, coefficients_A)

Terms a(82) onward added by G. C. Greubel, Dec 10 2017

A346577 a(n) = (1/(3n)) Sum_{d|n} mu(n/d) * binomial(3*d,d).

Original entry on oeis.org

1, 2, 9, 40, 200, 1026, 5537, 30624, 173583, 1001400, 5864749, 34768296, 208267319, 1258574114, 7663720500, 46976003712, 289628805622, 1794932293950, 11175157356521, 69864074596000, 438403736543598, 2760351027094298, 17433869214973753, 110420300844952992

Offset: 1

Ilya Gutkovskiy, Jul 24 2021

nonn

Inverse Euler transform of A001764.

Moebius transform of A082936.

Cf. A001764, A005809, A008683, A022553, A060170, A082936, A346578, A346579, A346580, A346581, A346582.

Table[(1/(3 n)) Sum[MoebiusMu[n/d] Binomial[3 d, d], {d, Divisors[n]}], {n, 24}]

a(n) = sumdiv(n, d, moebius(n/d)*binomial(3*d,d))/(3*n); \\ Michel Marcus, Jul 24 2021

a(n) = A060170(n)/3. - Hugo Pfoertner, Jul 24 2021

A346578 a(n) = (1/(4n)) Sum_{d|n} mu(n/d) * binomial(4*d,d).

Original entry on oeis.org

1, 3, 18, 112, 775, 5598, 42287, 328640, 2615085, 21191125, 174303162, 1451424960, 12211799223, 103655906781, 886568152950, 7633233227520, 66105170315083, 575445689879247, 5032380942945321, 44191451767056400, 389514699012969936, 3444925385161998518, 30561576846316109863

Offset: 1

Ilya Gutkovskiy, Jul 24 2021

nonn

Inverse Euler transform of A002293.

Moebius transform of A261497.

Cf. A002293, A005810, A008683, A022553, A261497, A346577, A346579, A346580, A346581, A346582.

Table[(1/(4 n)) Sum[MoebiusMu[n/d] Binomial[4 d, d], {d, Divisors[n]}], {n, 23}]

a(n) = sumdiv(n, d, moebius(n/d)*binomial(4*d,d))/(4*n); \\ Michel Marcus, Jul 24 2021

A346579 a(n) = (1/(5n)) Sum_{d|n} mu(n/d) * binomial(5*d,d).

Original entry on oeis.org

1, 4, 30, 240, 2125, 19776, 192129, 1922496, 19692504, 205444500, 2175519379, 23322637440, 252631900235, 2760767859780, 30400169155500, 336977763170048, 3757141504436392, 42107201575798248, 474084628585822412, 5359833703935374000, 60823006052351537106, 692556314455384443196

Offset: 1

Ilya Gutkovskiy, Jul 24 2021

nonn

Inverse Euler transform of A002294.

Moebius transform of A261498.

Cf. A001449, A002294, A008683, A022553, A261498, A346577, A346578, A346580, A346581, A346582.

Table[(1/(5 n)) Sum[MoebiusMu[n/d] Binomial[5 d, d], {d, Divisors[n]}], {n, 22}]

a(n) = sumdiv(n, d, moebius(n/d)*binomial(5*d,d))/(5*n); \\ Michel Marcus, Jul 24 2021

A346580 a(n) = (1/(6n)) Sum_{d|n} mu(n/d) * binomial(6*d,d).

Original entry on oeis.org

1, 5, 45, 440, 4750, 54081, 642341, 7861216, 98480286, 1256564750, 16273981757, 213378921432, 2826867619108, 37782552518473, 508840821825750, 6898459208449920, 94070535317459017, 1289430373107917718, 17755914760643605781, 245518560759177014000, 3407586451859019939012

Offset: 1

Ilya Gutkovskiy, Jul 24 2021

nonn

Inverse Euler transform of A002295.

Moebius transform of A261499.

Cf. A002295, A004355, A008683, A022553, A261499, A346577, A346578, A346579, A346581, A346582.

Table[(1/(6 n)) Sum[MoebiusMu[n/d] Binomial[6 d, d], {d, Divisors[n]}], {n, 21}]

a(n) = sumdiv(n, d, moebius(n/d)*binomial(6*d,d))/(6*n); \\ Michel Marcus, Jul 24 2021

A346581 a(n) = (1/(7n)) Sum_{d|n} mu(n/d) * binomial(7*d,d).

Original entry on oeis.org

1, 6, 63, 728, 9275, 124866, 1753073, 25365600, 375677595, 5667202850, 86775157139, 1345153422600, 21069043965983, 332927798516614, 5301031234076325, 84967018610221440, 1369846562874360886, 22199151535757655354, 361411377745122110421, 5908312923789590118600

Offset: 1

Ilya Gutkovskiy, Jul 24 2021

nonn

Inverse Euler transform of A002296.

Moebius transform of A261500.

Cf. A002296, A004368, A008683, A022553, A261500, A346577, A346578, A346579, A346580, A346582.

Table[(1/(7 n)) Sum[MoebiusMu[n/d] Binomial[7 d, d], {d, Divisors[n]}], {n, 20}]

a(n) = sumdiv(n, d, moebius(n/d)*binomial(7*d,d))/(7*n); \\ Michel Marcus, Jul 24 2021

A346582 a(n) = (1/(8n)) Sum_{d|n} mu(n/d) * binomial(8*d,d).

Original entry on oeis.org

1, 7, 84, 1120, 16450, 255612, 4141382, 69158272, 1182125043, 20581143150, 363704640475, 6506965023168, 117626432708863, 2145180354493274, 39421026305266125, 729242353100281344, 13568988503585900647, 253785064585174078869, 4768543107831461199896, 89970814565326816488000

Offset: 1

Ilya Gutkovskiy, Jul 24 2021

nonn

Inverse Euler transform of A007556.

Moebius transform of A261501.

Cf. A004381, A007556, A008683, A022553, A261501, A346577, A346578, A346579, A346580, A346581.

Table[(1/(8 n)) Sum[MoebiusMu[n/d] Binomial[8 d, d], {d, Divisors[n]}], {n, 20}]

a(n) = sumdiv(n, d, moebius(n/d)*binomial(8*d,d))/(8*n); \\ Michel Marcus, Jul 24 2021

A005354 Number of asymmetric planar trees with n nodes.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 3, 9, 28, 85, 262, 827, 2651, 8626, 28507, 95393, 322938, 1104525, 3812367, 13266366, 46504495, 164098390, 582521687, 2079133141, 7457788295, 26872946466, 97238824018, 353218128299, 1287657977946, 4709784136316

Offset: 0

N. J. A. Sloane, Simon Plouffe, Susanna Cuyler

a(13) in the Labelle table is a typographical error. - R. J. Mathar, Feb 03 2010

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Vincenzo Librandi)
Gilbert Labelle, Counting asymmetric enriched trees, J. Symbolic Comput. 14 (1992), no. 2-3, 211-242.
Torsten Mütze and Franziska Weber, Construction of 2-factors in the middle layer of the discrete cube, arXiv preprint arXiv:1111.2413 [math.CO], 2011.
T. Mütze and F. Weber, Construction of 2-factors in the middle layer of the discrete cube, Journal of Combinatorial Theory, Series A, 119(8) (2012), 1832-1855.
Index entries for sequences related to trees

Cf. A000108, A002995, A022553.

From R. J. Mathar, Feb 03 2010: (Start)
A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A007727 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do a := a+binomial(2*d,d)*numtheory[mobius](n/d) ; end do ; a ; end proc;
A022553 := proc(n) A007727(n)/2/n ; end proc:
A005354 := proc(n) local a; if n <=1 then 1; else a := A022553(n-1) ; a := a-A000108(n-1)/2 ; if type(n,'even') then a := a-A000108(n/2-1)/2 ; end if; a ; end if; end proc: seq(A005354(n),n=0..20) ; (End)

a[0] = a[1] = 1; a[n_] := DivisorSum[n-1, MoebiusMu[(n-1)/#]*Binomial[2#, #]&]/(2(n-1)) - CatalanNumber[n-1]/2 - Boole[EvenQ[n]]*CatalanNumber[n/2 - 1]/2; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, May 09 2012, after R. J. Mathar, updated Jan 31 2018 *)

From Christian G. Bower, Dec 15 1999: (Start)
G.f.: 1+B(x)+(C(x^2)-C(x)^2)/2 where B is g.f. of A022553(n-1) and C is g.f. of A000108(n-1).
a(n) = A022553(n-1) - A000108(n-2)/2 - (if n is even) A000108(n/2-1)/2. (End)

More terms from Christian G. Bower, Dec 15 1999

cp's OEIS Frontend

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

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A268619 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)*binomial(2*d,d).

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A178904 This should be related to the Coxeter transformations of the posets of partitions in rectangular boxes of size m times n.

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A346577 a(n) = (1/(3*n)) * Sum_{d|n} mu(n/d) * binomial(3*d,d).

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A346578 a(n) = (1/(4*n)) * Sum_{d|n} mu(n/d) * binomial(4*d,d).

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A346579 a(n) = (1/(5*n)) * Sum_{d|n} mu(n/d) * binomial(5*d,d).

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A346580 a(n) = (1/(6*n)) * Sum_{d|n} mu(n/d) * binomial(6*d,d).

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A346581 a(n) = (1/(7*n)) * Sum_{d|n} mu(n/d) * binomial(7*d,d).

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

A268619 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)binomial(2d,d).

A346577 a(n) = (1/(3n)) Sum_{d|n} mu(n/d) * binomial(3*d,d).

A346578 a(n) = (1/(4n)) Sum_{d|n} mu(n/d) * binomial(4*d,d).

A346579 a(n) = (1/(5n)) Sum_{d|n} mu(n/d) * binomial(5*d,d).

A346580 a(n) = (1/(6n)) Sum_{d|n} mu(n/d) * binomial(6*d,d).

A346581 a(n) = (1/(7n)) Sum_{d|n} mu(n/d) * binomial(7*d,d).

A346582 a(n) = (1/(8n)) Sum_{d|n} mu(n/d) * binomial(8*d,d).