A135860 -id:A135860 - 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

A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 15, 1, 1, 4, 70, 220, 1, 1, 5, 210, 5005, 4845, 1, 1, 6, 495, 48620, 735471, 142506, 1, 1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1, 1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1, 1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1

Offset: 0

Paul D. Hanna, Sep 04 2013

Central coefficients are A201555(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.

			The triangle of coefficients C(n*k, k^2), n>=k, k=0..n, begins:
1;
1, 1;
1, 2, 1;
1, 3, 15, 1;
1, 4, 70, 220, 1;
1, 5, 210, 5005, 4845, 1;
1, 6, 495, 48620, 735471, 142506, 1;
1, 7, 1001, 293930, 30421755, 183579396, 5245786, 1;
1, 8, 1820, 1307504, 601080390, 40225345056, 69668534468, 231917400, 1;
1, 9, 3060, 4686825, 7307872110, 3169870830126, 96926348578605, 37387265592825, 11969016345, 1; ...

Paul D. Hanna, Rows 0..30, as a flattened table of n, a(n) for n = 0..495

Cf. A228808 (row sums), A228833 (antidiagonal sums), A135860 (diagonal), A201555 (central terms).
Cf. A229052.
Cf. related triangles: A228904 (exp), A209330, A226234, A228836.

{T(n, k)=binomial(n*k, k^2)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

A135862 a(n) = binomial(n*(n+1),n)/(n^2+1).

Original entry on oeis.org

1, 1, 3, 22, 285, 5481, 141778, 4638348, 184138713, 8612835715, 464333035881, 28368973183200, 1938034271677595, 146439782923866810, 12129248258088578100, 1092975597259714940696, 106463932364272178140209, 11148388021809242372111895, 1248977393391271550436645787

Offset: 0

Paul D. Hanna, Dec 02 2007

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 - n + 1)). Cf. A143669.
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.
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. A135860, A135861, A143669.

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

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

a(n)=binomial(n*(n+1),n)/(n^2+1)
for(n=0,30,print1(a(n),", "))

{a(n) = local(A=1+x); for(i=1, n, A = exp( sum(k=1, n, A^(n*k)*x^k/k +x*O(x^n)))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

a(n) = A135860(n)/(n^2+1).
a(n) = 1/(n^2 + n + 1)*binomial(n^2 + n + 1, n). - Peter Bala, Dec 02 2015
Ccongruences: a(p) == (1 + p - p^2 - p^3) ( mod p^4 ) for prime p >= 5 and a(2*p) == (1 + 6*p + 4*p^2) ( mod p^3 ) for all prime p (apply Mestrovic, equation 37). - Peter Bala, Feb 23 2020

A107863 Column 1 of triangle A107862; a(n) = binomial(n*(n+1)/2 + n, n).

Original entry on oeis.org

1, 2, 10, 84, 1001, 15504, 296010, 6724520, 177232627, 5317936260, 179013799328, 6681687099710, 273897571557780, 12233149001721760, 591315394579074378, 30756373941461374800, 1712879663609111933495, 101696990867999141755140

Offset: 0

Paul D. Hanna, Jun 04 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 0..350
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, A107862, A135860, A135861.

Table[Binomial[n*(n+3)/2, n], {n,0,40}] (* G. C. Greubel, Feb 19 2022 *)

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

[binomial(n*(n+3)/2, n) for n in (0..40)] # G. C. Greubel, Feb 19 2022

a(n) = [x^(n*(n+1)/2)] 1/(1 - x)^(n+1). - Ilya Gutkovskiy, Oct 10 2017
From Peter Bala, Feb 23 2020: (Start)
Put b(n) = a(n-1). We have the congruences:
b(p) == 1 (mod p^3) for prime p >= 5 (uses Mestrovic, equation 35);
b(2*p) == 2*p (mod p^4) for prime p >= 5 (uses Mestrovic, equation 44 and the von Staudt-Clausen theorem).
Conjectural congruences:
b(3*p) == (81*p*2 - 1)/8 (mod p^3) for prime p >= 3;
3*b(4*p) == -4*p (mod p^3) for all prime p. Cf. A135860 and A135861. (End)

A272238 a(n) = Product_{k=0..n} (n^2+k)!.

Original entry on oeis.org

1, 2, 2073600, 25177856146146034974720000000, 14100949826093501607549529280892932893801777581548587107609477120000000000000000

Offset: 0

Vaclav Kotesovec, Apr 23 2016

The next term has 173 digits.

Cf. A135860, A272164, A272180, A272237, A272241.

Table[Product[(n^2+k)!, {k, 0, n}], {n, 0, 6}]
Table[BarnesG[n^2 + n + 2]/BarnesG[n^2 + 1], {n, 0, 6}]

a(n) = ((n^2+n)!)^(n+1) / A272237(n).

a(n) ~ exp(11/24 + n/6 - n^2 - n^3) * n^((1+n)*(1 + n + 2*n^2)) * (2*Pi)^((n+1)/2).

A295763 G.f. satisfies: A(x) = Sum_{n>=0} binomial(n(n+1),n)/(n+1) x^n/A(x)^n.

Original entry on oeis.org

1, 1, 4, 42, 744, 18570, 596929, 23457763, 1089601420, 58424516424, 3553205095552, 241765128267597, 18202737707568180, 1502857964050898494, 135033771405912550765, 13119213786776385900734, 1370572549521961522812200, 153224265349198540163190599, 18253426026439076436840194131, 2308479498698233016622014842489

Offset: 0

Paul D. Hanna, Jan 06 2018

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 42*x^3 + 744*x^4 + 18570*x^5 + 596929*x^6 + 23457763*x^7 + 1089601420*x^8 + 58424516424*x^9 + 3553205095552*x^10 +...
such that
A(x) = 1 + x/A(x) + 5*(x/A(x))^2 + 55*(x/A(x))^3 + 969*(x/A(x))^4 + 23751*(x/A(x))^5 + 749398*(x/A(x))^6 +...+ binomial(n*(n+1),n)/(n+1)*(x/A(x))^n +...
The table of coefficients of x^k in A(x)^(n+1) begins:
  [1, 1, 4, 42, 744, 18570, 596929, 23457763, 1089601420, ...];
  [1, 2, 9, 92, 1588, 38964, 1238714, 48320440, 2233007214, ...];
  [1, 3, 15, 151, 2544, 61356, 1928659, 74668905, 3432698217, ...];
  [1, 4, 22, 220, 3625, 85936, 2670332, 102589280, 4691284160, ...];
  [1, 5, 30, 300, 4845, 112911, 3467585, 132173305, 6011511390, ...];
  [1, 6, 39, 392, 6219, 142506, 4324575, 163518732, 7396271082, ...];
  [1, 7, 49, 497, 7763, 174965, 5245786, 196729744, 8848607971, ...];
  [1, 8, 60, 616, 9494, 210552, 6236052, 231917400, 10371729633, ...];
  [1, 9, 72, 750, 11430, 249552, 7300581, 269200107, 11969016345, ...]; ...
in which the main diagonal begins:
  [1, 2, 15, 220, 4845, 142506, 5245786, ..., binomial(n*(n+1),n), ...],
thus [x^n] A(x)^(n+1) = [x^n] (1 + x)^(n*(n+1)) for n>=0.

Paul D. Hanna, Table of n, a(n) for n = 0..300

cf. A295764, A295765, A135861, A135860.

terms = 20; A[] = 1; Do[A[x] = Sum[Binomial[n*(n+1), n]/(n+1)*x^n/A[x]^n, {n, 0, terms}] + O[x]^terms // Normal, terms];
CoefficientList[A[x], x] (* Jean-François Alcover, Jan 14 2018 *)

{a(n) = my(A=[1]); for(m=1,n, A = concat(A,0); V = Vec( Ser(A)^(m+1) ); A[m+1] = (binomial(m*(m+1),m) - V[m+1])/(m+1);); A[n+1]}
for(n=0,20,print1(a(n),", "))

G.f. A(x) satisfies: [x^n] A(x)^(n+1) = binomial(n*(n+1),n) for n>=0.

a(n) ~ c * exp(n) * n^(n - 3/2), where c = exp(1/2 - exp(-2)) / sqrt(2*Pi) = 0.5744892944370457395619... - Vaclav Kotesovec, Oct 17 2020, updated Mar 18 2024

A306280 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(n^2+k,k).

Original entry on oeis.org

1, 3, 26, 416, 9708, 297662, 11306572, 512307336, 26968496504, 1617489748394, 108885682104744, 8129721925098468, 666736347200187804, 59582961423951290184, 5762936296492591067968, 599807329803134064385488, 66843498592187788579795440

Offset: 0

Seiichi Manyama, Feb 02 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..337

Cf. A001850, A114496, A135860, A156886, A156887.

a[n_] := Sum[Binomial[n,k] * Binomial[n^2+k,k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 03 2019 *)

{a(n) = sum(k=0, n, binomial(n, k)*binomial(n^2+k, k))}

From Vaclav Kotesovec, Feb 08 2019: (Start)
a(n) ~ exp(1) * A135860(n).
a(n) ~ exp(n + 3/2) * n^(n - 1/2) / sqrt(2*Pi). (End)

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).

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

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A228832 Triangle defined by T(n,k) = binomial(n*k, k^2), for n>=0, k=0..n, as read by rows.

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A135862 a(n) = binomial(n*(n+1),n)/(n^2+1).

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A107863 Column 1 of triangle A107862; a(n) = binomial(n*(n+1)/2 + n, n).

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A272238 a(n) = Product_{k=0..n} (n^2+k)!.

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A295763 G.f. satisfies: A(x) = Sum_{n>=0} binomial(n*(n+1),n)/(n+1) * x^n/A(x)^n.

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A295763 G.f. satisfies: A(x) = Sum_{n>=0} binomial(n(n+1),n)/(n+1) x^n/A(x)^n.