A135862 -id:A135862 - cp's OEIS frontend

A014068 a(n) = binomial(n*(n+1)/2, n).

1, 1, 3, 20, 210, 3003, 54264, 1184040, 30260340, 886163135, 29248649430, 1074082795968, 43430966148115, 1917283000904460, 91748617512913200, 4730523156632595024, 261429178502421685800, 15415916972482007401455, 966121413245991846673830, 64123483527473864490450300

Offset: 0

N. J. A. Sloane

nonn

Product of next n numbers divided by product of first n numbers. E.g., a(4) = (7*8*9*10)/(1*2*3*4)= 210. - Amarnath Murthy, Mar 22 2004

Also the number of labeled loop-graphs with n vertices and n edges. The covering case is A368597. - Gus Wiseman, Jan 25 2024

			From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(3) = 20 loop-graph edge-sets (loops shown as singletons):
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
             {{1},{1,2}}  {{1},{2},{1,2}}
             {{2},{1,2}}  {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{1,3}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{2},{3},{2,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}
(End)

Harvey P. Dale, Table of n, a(n) for n = 0..370
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A000217, A022915, A135860, A135861, A135862.
Diagonal of A084546.
Without loops we have A116508, covering A367863, unlabeled A006649.
Allowing edges of any positive size gives A136556, covering A054780.
The covering case is A368597.
The unlabeled version is A368598, covering A368599.
The connected case is A368951.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 (shifted left) counts loop-graphs, covering A322661.
A006129 counts covering simple graphs, connected A001187.
A058891 counts set-systems, unlabeled A000612.
Cf. A000085, A000272, A057500, A333331, A368596, A368730.

[Binomial(Binomial(n+1,2), n): n in [0..40]]; // G. C. Greubel, Feb 19 2022

Binomial[First[#],Last[#]]&/@With[{nn=20},Thread[{Accumulate[ Range[ 0,nn]], Range[ 0,nn]}]] (* Harvey P. Dale, May 27 2014 *)

from math import comb
def A014068(n): return comb(comb(n+1,2),n) # Chai Wah Wu, Jul 14 2024

[(binomial(binomial(n+1, n-1), n)) for n in range(20)] # Zerinvary Lajos, Nov 30 2009

For n >= 1, Product_{k=1..n} a(k) = A022915(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
For n > 0, a(n) = A022915(n)/A022915(n-1). - Gerald McGarvey, Jul 26 2004
a(n) = binomial(T(n+1), T(n)) where T(n) = the n-th triangular number. - Amarnath Murthy, Jul 14 2005
a(n) = binomial(binomial(n+2, n), n+1) for n >= -1. - Zerinvary Lajos, Nov 30 2009
From Peter Bala, Feb 27 2020: (Start)
a(p) == (p + 1)/2 ( mod p^3 ) for prime p >= 5 (apply Mestrovic, equation 37).
Conjectural: a(2*p) == p*(2*p + 1) ( mod p^4 ) for prime p >= 5. (End)
a(n) = A084546(n,n). - Gus Wiseman, Jan 25 2024
a(n) = [x^n] (1+x)^(n*(n+1)/2). - Vaclav Kotesovec, Aug 06 2025

A135861 a(n) = binomial(n*(n+1),n)/(n+1).

Original entry on oeis.org

1, 1, 5, 55, 969, 23751, 749398, 28989675, 1329890705, 70625252863, 4263421511271, 288417894029200, 21616536107173175, 1778197364075525550, 159297460456229992380, 15438280311293473537331, 1609484153977526457766689, 179612918129148904884024975

Offset: 0

Paul D. Hanna, Dec 02 2007

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..338
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A014068, A107863, A135860, A135862, A295763.

A135861:=n->binomial(n*(n+1),n)/(n+1); seq(A135861(n), n=0..15); # Wesley Ivan Hurt, May 08 2014

Table[Binomial[n*(n + 1), n]/(n + 1), {n, 0, 15}]

PARI
```
a(n)=binomial(n*(n+1),n)/(n+1)
```

a(n) = A135860(n)/(n+1).
a(n) = [x^(n^2)] 1/(1 - x)^n. - Ilya Gutkovskiy, Oct 10 2017
a(p) == 1 ( mod p^4 ) for prime p >= 5 and a(2*p) == 4*p + 1 ( mod p^4 ) for prime p >= 5 (apply Mestrovic, equation 37). - Peter Bala, Feb 23 2020
a(n) ~ exp(n + 1/2) * n^(n - 3/2) / sqrt(2*Pi). - Vaclav Kotesovec, Oct 17 2020

A135860 a(n) = binomial(n*(n+1), n).

Original entry on oeis.org

1, 2, 15, 220, 4845, 142506, 5245786, 231917400, 11969016345, 706252528630, 46897636623981, 3461014728350400, 281014969393251275, 24894763097057357700, 2389461906843449885700, 247012484980695576597296, 27361230617617949782033713, 3233032526324680287912449550

Offset: 0

Paul D. Hanna, Dec 02 2007

Seiichi Manyama, Table of n, a(n) for n = 0..337
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A014062, A014068, A054688, A107863, A135861, A135862.

[Binomial(n*(n+1), n): n in [0..30]]; // G. C. Greubel, Feb 20 2022

Table[Binomial[n^2 + n, n], {n, 0, 16}] (* Arkadiusz Wesolowski, Jul 18 2012 *)
(* or *)
Table[SeriesCoefficient[(1+x)^(n*(n+1)), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

a(n)=binomial(n*(n+1),n)
for(n=0,15,print1(a(n),", "))

a(n)=sum(k=0,n,binomial(n,k)*binomial(n^2,k))
for(n=0,15,print1(a(n),", "))

[binomial(n*(n+1), n) for n in (0..30)] # G. C. Greubel, Feb 20 2022

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k). - Paul D. Hanna, Nov 18 2015
a(n) is divisible by (n+1): a(n)/(n+1) = A135861(n).
a(n) is divisible by (n^2+1): a(n)/(n^2+1) = A135862(n).
a(n) = binomial(2*A000217(n),n). - Arkadiusz Wesolowski, Jul 18 2012
a(n) = [x^n] 1/(1 - x)^(n^2+1). - Ilya Gutkovskiy, Oct 03 2017
a(n) ~ exp(n + 1/2) * n^(n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Feb 08 2019
a(p) == p + 1 ( mod p^4 ) for prime p >= 5 and a(2*p) == (4*p + 1)*(2*p + 1) ( mod p^4 ) for all prime p. Apply Mestrovic, equation 37. - Peter Bala, Feb 27 2020
a(n) = ((n^2 + n)!)/((n^2)! * n!). - Peter Luschny, Feb 27 2020
a(n) = [x^n] (1 + x)^(n*(n+1)). - Vaclav Kotesovec, Aug 06 2025

A143669 a(n) = binomial((n+1)^2, n) / (n+1)^2.

Original entry on oeis.org

1, 1, 4, 35, 506, 10472, 285384, 9706503, 397089550, 19022318084, 1045659267016, 64924369564353, 4496010926381352, 343688726144945040, 28753733905585301136, 2613784129155164386575, 256569498889138342791510, 27050758656206146528056236

Offset: 0

Paul D. Hanna, Aug 28 2008

From Peter Bala, Dec 02 2015: (Start)
Let x = p/q be a positive rational in reduced form with p,q > 0. Define Cat(x) = 1/(2*p + q)*binomial(2*p + q, p). Then Cat(n) = Catalan(n). This sequence is Cat(n/(n^2 + 1)). Cf. A135862.
See Armstrong et al. for combinatorial interpretations of these generalized Catalan number sequences. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..339
D. Armstrong, B. Rhoades, and N. Williams, Rational associahedra and noncrossing partitions arxiv:1305.7286v1 [math.CO], 2013.

Cf. A295765, A135862, A182316.

[Binomial((n+1)^2, n) / (n+1)^2: n in [0..20]]; // Vincenzo Librandi, Dec 09 2015

Table[Binomial[(n + 1)^2, n]/(n + 1)^2, {n, 0, 30}] (* Vincenzo Librandi, Dec 09 2015 *)

PARI
```
a(n)=binomial((n+1)^2,n)/(n+1)^2
```

A228509 a(n) = binomial(n^2+n+1,n) * (n+1) / (n^2+n+1) for n>=0.

Original entry on oeis.org

1, 2, 9, 88, 1425, 32886, 992446, 37106784, 1657248417, 86128357150, 5107663394691, 340427678198400, 25194445531808735, 2050156960934135340, 181938723871328671500, 17487609556155439051136, 1809886850192627028383553, 200670984392566362698014110, 23730570474434159458296269953

Offset: 0

Paul D. Hanna, Aug 23 2013

nonn

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 88*x^3 + 1425*x^4 + 32886*x^5 +...

Cf. A135862, A135860.

Table[(Binomial[n^2+n+1,n](n+1))/(n^2+n+1),{n,0,20}] (* Harvey P. Dale, Oct 13 2015 *)

{a(n)=binomial(n^2+n+1,n)*(n+1)/(n^2+n+1)}
for(n=0,20,print1(a(n),", "))

a(n) = (n+1)*A135862(n).

cp's OEIS Frontend

A014068 a(n) = binomial(n*(n+1)/2, n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Python

Sage

Formula

A135861 a(n) = binomial(n*(n+1),n)/(n+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A135860 a(n) = binomial(n*(n+1), n).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A143669 a(n) = binomial((n+1)^2, n) / (n+1)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A228509 a(n) = binomial(n^2+n+1,n) * (n+1) / (n^2+n+1) for n>=0.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula