A135861 -id:A135861 - 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

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

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)

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

A298692 G.f. A(x) satisfies: 1 = Sum_{n>=0} binomial( n(n+1), n)/(n+1) x^n / A(x)^( (n+1)^2 ).

Original entry on oeis.org

1, 1, 2, 15, 213, 4485, 123566, 4171778, 166069875, 7602292250, 393220294258, 22679300697606, 1443478702575162, 100529312696403699, 7606562231567559478, 621526322941129712986, 54553240678513466719077, 5120001583257750960650134, 511729676123794537164792892, 54270040973557127212080028474, 6087267497390906756985330494931

Offset: 0

Paul D. Hanna, Feb 03 2018

nonn

Compare to: 1 = Sum_{n>=0} binomial(m*(n+1), n)/(n+1) * x^n / (1+x)^(m*(n+1)) holds for fixed m.

			G.f.: A(x) = 1 + x + 2*x^2 + 15*x^3 + 213*x^4 + 4485*x^5 + 123566*x^6 + 4171778*x^7 + 166069875*x^8 + 7602292250*x^9 + 393220294258*x^10 + 22679300697606*x^11 + 1443478702575162*x^12 + 100529312696403699*x^13 + 7606562231567559478*x^14 + 621526322941129712986*x^15 + ...
such that
1 = 1/A(x) + C(2,1)/2*x/A(x)^4 + C(6,2)/3*x^2/A(x)^9 + C(12,3)/4*x^3/A(x)^16 + C(20,4)/5*x^4/A(x)^25 + C(30,5)/6*x^5/A(x)^36 + C(42,6)/7*x^6/A(x)^49 + C(56,7)/8*x^7/A(x)^64 + ...
more explicitly,
1 = 1/A(x) + x/A(x)^4 + 5*x^2/A(x)^9 + 55*x^3/A(x)^16 + 969*x^4/A(x)^25 + 23751*x^5/A(x)^36 + 749398*x^6/A(x)^49 + 28989675*x^7/A(x)^64 + ... + A135861(n)*x^n/A(x)^((n+1)^2) + ...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A135861.

terms = 21; A[] = 1; Do[A[x] = A[x] - 1 + Sum[Binomial[n*(n+1), n]/(n+1)*x^n/A[x]^((n + 1)^2) + O[x]^(terms), {n, 0, k}], {k, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Feb 06 2018 *)

{a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m*(m+1),m)/(m+1) * x^m/Ser(A)^((m+1)^2-1) ))); A[n+1]}
for(n=0,20,print1(a(n),", "))

a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.17655460948348035823168... and c = 0.16989702... - Vaclav Kotesovec, Oct 17 2020

A298696 G.f.: Sum_{n>=0} binomial(n(n+1), n)/(n+1) x^n / (1 + x)^(n*(n+1)).

Original entry on oeis.org

1, 1, 3, 28, 410, 8386, 220962, 7140736, 273712896, 12146997564, 612813677300, 34647736132384, 2170381958609592, 149223874286440552, 11173356309069883320, 905099760309260722560, 78870011549256151244288, 7356892186010414244194704, 731435433368215011644979504, 77216368897429504869064200256, 8626428901029156775683110378400, 1016792561657783042048699052986016

Offset: 0

Paul D. Hanna, Feb 04 2018

nonn

Compare g.f. to: 1 = Sum_{n>=0} binomial(m*(n+1), n)/(n+1) * x^n / (1+x)^(m*(n+1)) holds for fixed m.

			G.f.: A(x) = 1 + x + 3*x^2 + 28*x^3 + 410*x^4 + 8386*x^5 + 220962*x^6 + 7140736*x^7 + 273712896*x^8 + 12146997564*x^9 + 612813677300*x^10 + ...
such that
A(x) = 1 + C(2,1)/2*x/(1+x)^2 + C(6,2)/3*x^2/(1+x)^6 + C(12,3)/4*x^3/(1+x)^12 + C(20,4)/5*x^4/(1+x)^20 + C(30,5)/6*x^5/(1+x)^30 + ...
more explicitly,
A(x) = 1 + x/(1+x)^2 + 5*x^2/(1+x)^6 + 55*x^3/(1+x)^12 + 969*x^4/(1+x)^20 + 23751*x^5/(1+x)^30 + ... + A135861(n)*x^n/(1+x)^(n*(n+1)) + ...

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

Cf. A298695, A135861.

terms = 22; s = Sum[Binomial[n*(n + 1), n]/(n + 1)*x^n/(1 + x)^(n*(n + 1)), {n, 0, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Feb 05 2018 *)

{a(n) = my(A = sum(m=0,n,binomial(m*(m+1),m)/(m+1)*x^m/(1+x +x*O(x^n))^(m*(m+1)) ) ); polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))

a(n) ~ c * d^n * n! / n^2, where d = -4 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 6.176554609483480358231680164050876553672889... and c = 0.226094037474708064867716267720651240574569526310006844420310030408773601638... - Vaclav Kotesovec, Feb 07 2018

c = exp(LambertW(-2*exp(-2))^2/8 - 1/2) * sqrt(2) * (2 + LambertW(-2*exp(-2))) / (4*Pi*sqrt(-LambertW(-2*exp(-2))) * sqrt(1 + LambertW(-2*exp(-2)))). - Vaclav Kotesovec, Mar 18 2022

A366012 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n*(n+1),n-k).

Original entry on oeis.org

1, 2, 13, 156, 2833, 70098, 2214280, 85464984, 3906724321, 206648387550, 12425282899588, 837384222603448, 62539219710804627, 5127758187193514824, 457986530357734020432, 44263628968974498793648, 4602969726808566383149761, 512486177498084438210961270, 60827938291895363867587959628

Offset: 0

Ilya Gutkovskiy, Sep 25 2023

nonn

Main diagonal of A107111.

Cf. A000108, A006318, A062992, A135861, A263843, A365754, A365755.

Table[1/(n + 1) Sum[Binomial[n + k, n] Binomial[n (n + 1), n - k], {k, 0, n}], {n, 0, 18}]
Table[SeriesCoefficient[(1/x) InverseSeries[Series[x (1 - x)/(1 + x)^n, {x, 0, n + 1}], x], {x, 0, n}], {n, 0, 18}]

a(n) = [x^n] (1/x) * Series_Reversion( x * (1 - x) / (1 + x)^n ).

a(n) ~ exp(n + 3/2) * n^(n - 3/2) / sqrt(2*Pi). - Vaclav Kotesovec, Sep 26 2023

A298693 G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n(n+1), n)/(n+1) x^n / A(x)^( n*(n+1) ).

Original entry on oeis.org

1, 1, 3, 22, 294, 5911, 158293, 5251690, 206696194, 9387611937, 482745371458, 27717788095397, 1757818683339028, 122058148921357056, 9212494564360610855, 751138761646263512978, 65807775099574132000968, 6166278653572358495161057, 615421469545011786309942067, 65183859793912213778457542207, 7303117991652113167690085149033

Offset: 0

Paul D. Hanna, Feb 03 2018

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 22*x^3 + 294*x^4 + 5911*x^5 + 158293*x^6 + 5251690*x^7 + 206696194*x^8 + 9387611937*x^9 + 482745371458*x^10 + 27717788095397*x^11 + 1757818683339028*x^12 + 122058148921357056*x^13 + 9212494564360610855*x^14 + 751138761646263512978*x^15 + ...
such that
A(x) = 1 + C(2,1)/2*x/A(x)^2 + C(6,2)/3*x^2/A(x)^6 + C(12,3)/4*x^3/A(x)^12 + C(20,4)/5*x^4/A(x)^20 + C(30,5)/6*x^5/A(x)^30 + C(42,6)/7*x^6/A(x)^42 + C(56,7)/8*x^7/A(x)^56 + ...
more explicitly,
A(x) = 1 + x/A(x)^2 + 5*x^2/A(x)^6 + 55*x^3/A(x)^12 + 969*x^4/A(x)^20 + 23751*x^5/A(x)^30 + 749398*x^6/A(x)^42 + 28989675*x^7/A(x)^56 + ... + A135861(n)*x^n/A(x)^(n*(n+1)) + ...

Cf. A298692, A135861.

terms = 21; A[] = 1; Do[A[x] = 1 + Sum[Binomial[n*(n+1), n]/(n+1)*x^n/ A[x]^(n*(n+1)), {n, terms}] + O[x]^terms, {terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Feb 09 2018 *)

{a(n) = my(A=[1]); for(i=1,n, A = Vec(sum(m=0,#A,binomial(m*(m+1),m)/(m+1) * x^m/Ser(A)^(m*(m+1)) )));  A[n+1]}
for(n=0,20,print1(a(n),", "))

A366038 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n(n+1),n-k) n^k.

Original entry on oeis.org

1, 2, 25, 658, 27193, 1548526, 112916830, 10062563610, 1061196371665, 129369938790070, 17909387604206371, 2776290021986848588, 476539253976442601735, 89736215305419802692184, 18395742890606906720656524, 4078527943680251523126851306, 972490249766494185823234587681

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Cf. A006318, A064063, A135861, A291699, A292798, A365816, A366012, A366016, A366037.

A366038 := proc(n)
    add(binomial(n+k,k)*binomial(n*(n+1),n-k)*n^k,k=0..n) ;
    %/(n+1) ;
end proc:
seq(A366038(n),n=0..80) ; # R. J. Mathar, Oct 24 2024

Unprotect[Power]; 0^0 = 1; Table[1/(n + 1) Sum[Binomial[n + k, k] Binomial[n (n + 1) , n - k] n^k, {k, 0, n}], {n, 0, 16}]
Table[Binomial[n (n + 1), n] Hypergeometric2F1[-n, n + 1, n^2 + 1, -n]/(n + 1), {n, 0, 16}]
Table[SeriesCoefficient[(1/x) InverseSeries[Series[x (1 - n x)/(1 + x)^n, {x, 0, n + 1}], x], {x, 0, n}], {n, 0, 16}]

a(n) = [x^n] (1/x) * Series_Reversion( x * (1 - n * x) / (1 + x)^n ).

a(n) ~ phi^(3*n + 3/2) * exp(n/phi^2 + 1/(2*phi)) * n^(n - 3/2) / (5^(1/4) * sqrt(2*Pi)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Sep 27 2023

cp's OEIS Frontend

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

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

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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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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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A298692 G.f. A(x) satisfies: 1 = Sum_{n>=0} binomial( n*(n+1), n)/(n+1) * x^n / A(x)^( (n+1)^2 ).

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A298696 G.f.: Sum_{n>=0} binomial(n*(n+1), n)/(n+1) * x^n / (1 + x)^(n*(n+1)).

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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.

A298692 G.f. A(x) satisfies: 1 = Sum_{n>=0} binomial( n(n+1), n)/(n+1) x^n / A(x)^( (n+1)^2 ).

A298696 G.f.: Sum_{n>=0} binomial(n(n+1), n)/(n+1) x^n / (1 + x)^(n*(n+1)).

A298693 G.f. A(x) satisfies: A(x) = Sum_{n>=0} binomial( n(n+1), n)/(n+1) x^n / A(x)^( n*(n+1) ).

A366038 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n(n+1),n-k) n^k.