A014062 -id:A014062 - cp's OEIS frontend

A383120 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k).

1, 2, 11, 139, 2885, 82381, 2979565, 130203494, 6664589321, 390857822425, 25832193906761, 1899273577364197, 153741850998047053, 13585520026454056279, 1301210398133681268381, 134270617908678099820891, 14849785991790603714043921, 1752283118795349858851381297

Offset: 0

Ilya Gutkovskiy, Apr 17 2025

nonn

Cf. A001850, A014062, A026375, A188686, A226391, A359643, A378327, A383121.

Table[Sum[Binomial[n, k] Binomial[n k, k], {k, 0, n}], {n, 0, 17}]

a(n) = sum(k=0, n, binomial(n,k) * binomial(n*k,k)); \\ Michel Marcus, Apr 17 2025

a(n) = [x^n] ((1 + x)^n + x)^n.

a(n) ~ exp(n + exp(-1) - 1/2) * n^n / sqrt(2*Pi*n). - Vaclav Kotesovec, Apr 17 2025

A055789 a(n) = binomial(n, round(sqrt(n))).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 15, 35, 56, 84, 120, 165, 220, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 736281, 906192, 1107568, 1344904, 1623160, 1947792, 2324784, 2760681

Offset: 0

Henry Bottomley, Jul 13 2000

			a(9) = C(9,3) = 9!/(3!*6!) = 84

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000194, A014062.

[Binomial(n, Round(Sqrt(n))): n in [0..40]]; // G. C. Greubel, Jan 25 2020

seq( binomial(n, round(sqrt(n))), n=0..40); # G. C. Greubel, Jan 25 2020

Table[Binomial[n, Round[Sqrt[n]]], {n,0,40}] (* G. C. Greubel, Jan 25 2020 *)

vector(40, n, binomial(n, round(sqrt(n))) ) \\ G. C. Greubel, Jan 25 2020

from math import comb, isqrt
def A055789(n): return comb(n,(m:=isqrt(n))+ int((n-m*(m+1)<<2)>=1)) # Chai Wah Wu, Jul 29 2022

[binomial(n, round(sqrt(n))) for n in (0..40)] # G. C. Greubel, Jan 25 2020

a(n^2) = A014062(n).

A066789 a(n) = binomial(n^3, n^2).

Original entry on oeis.org

1, 70, 4686825, 488526937079580, 130054841538480192455912505, 134084922435426494254000700271928170048684, 759782631286509488749614088922952734321921774350851547360350

Offset: 1

Benoit Cloitre, Jan 18 2002

nonn

Harry J. Smith, Table of n, a(n) for n = 1..23

Cf. A014062 = C(n^2, n).

Table[Binomial[n^3, n^2], {n, 8}] (* Zak Seidov, May 26 2005 *)

a(n) = { binomial(n^3, n^2) } \\ Harry J. Smith, Mar 26 2010

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007

A107447 a(n) = binomial(n!, n^2).

Original entry on oeis.org

1, 0, 0, 735471, 41749257038001257014137504, 8072776113194557737391130747136885454937928869204466648295480, 34145180671088019813488798475366394184193477615213303683031012996650190080826664983024305988320017979711374101114480000

Offset: 1

Zak Seidov, May 26 2005

G. C. Greubel, Table of n, a(n) for n = 1..11

Cf. A014062 C(n^2, n), A086687 C(n!, n), A067454.

[Binomial(Factorial(n), n^2): n in [1..9]]; // G. C. Greubel, Mar 24 2024

Mathematica
```
a[n_]:= Binomial[n!, n^2]; Array[a, 10]
```

[binomial(factorial(n), n^2) for n in range(1,10)] # G. C. Greubel, Mar 24 2024

From Vaclav Kotesovec, May 28 2025: (Start)
a(n) ~ n!^(n^2) / (n^2)!.
a(n) ~ (2*Pi)^((n^2 - 1)/2) * n^(n^3 - 3*n^2/2 - 1) / exp(n^3 -n^2 - n/12). (End)

A177424 Exponent of the highest power of 2 dividing binomial(n^2,n).

Original entry on oeis.org

0, 0, 1, 2, 2, 1, 4, 3, 3, 1, 3, 3, 2, 3, 5, 4, 4, 1, 3, 3, 5, 1, 4, 8, 3, 2, 3, 5, 6, 4, 6, 5, 5, 1, 3, 3, 5, 2, 6, 3, 3, 3, 4, 3, 4, 4, 5, 6, 4, 2, 3, 7, 5, 1, 5, 5, 3, 4, 6, 6, 7, 5, 7, 6, 6, 1, 3, 3, 5, 2, 6, 4, 7, 1, 3, 3, 3, 5, 6, 4, 4, 2, 6, 3, 5, 5, 4, 6, 6, 2, 4, 12, 6, 5, 6, 7, 5, 2, 3, 6, 5, 3, 6, 3, 6

Offset: 0

Michel Lagneau, May 07 2010

a(n) is the largest integer such that 2^a(n) divides binomial(n^2,n)=A014062(n).

a(n) is the number of carries when adding n to n^2-n in base 2. - Robert Israel, Oct 23 2019

			For n = 6, binomial(36,6) = 1947792 = 2^4*3*7*11*17*31, the highest power of 2 is 2^4, and the exponent of 2^4 is a(6)=4.

Robert Israel, Table of n, a(n) for n = 0..10000(n=0 to 1000 from Harvey P. Dale)
Wikipedia, Kummer's theorem

Cf. A014062, A000120, A177234

A007814 := proc(n) if type(n,'odd') then 0; else for p in ifactors(n)[2] do if op(1,p) = 2 then return op(2,p); end if; end do: end if; end proc:
A014062 := proc(n) binomial(n^2,n) ; end proc:
A177424 := proc(n) A007814(A014062(n)) ; end proc: seq(A177424(n),n=0..80) ;
# Alternative:
nc:= proc(a,b,c)
  local t;
  if c=0 and (a=0 or b=0) then return 0 fi;
  t:= (a mod 2) + (b mod 2) + c;
  if t < 2 then  procname(floor(a/2),floor(b/2),0)
  else  1 + procname(floor(a/2),floor(b/2),1)
  fi
end proc:
seq(nc(n,n^2-n,0),n=0..100); # Robert Israel, Oct 23 2019

IntegerExponent[Table[Binomial[n^2,n],{n,0,120}],2] (* Harvey P. Dale, Mar 31 2019 *)

valp(n,p=2)=my(s); while(n\=p, s+=n); s
a(n)=valp(n^2)-valp(n^2-n)-valp(n) \\ Charles R Greathouse IV, Jul 08 2022

from math import comb
def A177424(n): return (~(m:=comb(n**2,n))& m-1).bit_length() # Chai Wah Wu, Jul 08 2022

a(n) = A007814(A014062(n)).

Maple program replaced by a structured general version - R. J. Mathar, May 10 2010

A178842 a(n) = binomial((n-1)^2, n).

Original entry on oeis.org

0, 0, 4, 126, 4368, 177100, 8347680, 450978066, 27540584512, 1878392407320, 141629804643600, 11703541346076580, 1052134368066259632, 102250849636865496528, 10683770265451303535424, 1194448077521704400002650, 142288257910903254700704000, 17993390003427864738863790640

Offset: 1

Thomas Young, Jun 17 2010

nonn

Number of ways to place n objects in an (n-1) X (n-1) array (e.g., the number of ways to arrange stars in a flag's field pattern).

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A014062 (binomial(n^2, n)).

List([1..20], n -> Binomial((n-1)^2, n)); # G. C. Greubel, Jan 21 2019

[Binomial((n-1)^2,n): n in [1..20]]; // G. C. Greubel, Jan 21 2019

a[n_] := Binomial[(n - 1)^2, n]; Array[a, 18] (* Robert G. Wilson v, Jul 25 2010 *)

vector(20, n, binomial((n-1)^2,n)) \\ G. C. Greubel, Jan 21 2019

[binomial((n-1)^2,n) for n in (1..20)] # G. C. Greubel, Jan 21 2019

A184357 a(n) = Sum_{k=0..n} C(n^2-k^2, n-k)*C(k^2, k).

Original entry on oeis.org

1, 2, 15, 226, 5079, 151326, 5611906, 248995090, 12862665297, 758353907422, 50255751919386, 3698524145800452, 299324750430958973, 26424096787968560864, 2527130527406877225450, 260305991718814269022586, 28732428200125730917353569

Offset: 0

Paul D. Hanna, Jan 15 2011

nonn

			a(0) = 1 = 1*1;
a(1) = 2 = 1*1 + 1*1;
a(2) = 15 = 6*1 + 3*1 + 1*6;
a(3) = 226 = 84*1 + 28*1 + 5*6 + 1*84;
a(4) = 5079 = 1820*1 + 455*1 + 66*6 + 7*84 + 1*1820;
a(5) = 151326 = 53130*1 + 10626*1 + 1330*6 + 120*84 + 9*1820 + 1*53130; ...

Harvey P. Dale, Table of n, a(n) for n = 0..337

Cf. A060539, A014062.

Table[Sum[Binomial[n^2-k^2,n-k]Binomial[k^2,k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jul 30 2023 *)

{a(n)=if(n<0, 0, sum(k=0, n, binomial(n^2-k^2, n-k)*binomial(k^2, k)))}

A186245 a(n) = binomial(n^2, 2*n).

Original entry on oeis.org

1, 84, 12870, 3268760, 1251677700, 675248872536, 488526937079580, 456703981505085600, 535983370403809682970, 771629762387959506903300, 1337261858854117218288723600, 2746379593275133584459064976784, 6596095888094645606758451183394760

Offset: 2

Alex Ratushnyak, Aug 18 2012

nonn

Michael De Vlieger, Table of n, a(n) for n = 2..205

Cf. A014062.

Array[Binomial[#^2, 2 #] &, 13, 2] (* Michael De Vlieger, May 21 2021 *)

import math
for n in range(2,22):
    x = n*n
    y = n+n
    print(math.factorial(x) / (math.factorial(x-y) * math.factorial(y)), end=',')

A187083 a(n) = binomial(n^n, n).

Original entry on oeis.org

1, 6, 2925, 174792640, 2475588476563125, 14320984850603177651837856, 50975600425441237253196072020826978589, 155681826868802708662507744652859497547627180714885120, 541851389452483826218851027234763464912884507272826833630475746754951097

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 03 2011

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..27
Harlan J. Brothers, Pascal's triangle, Sidi polynomials, and powers of e, Missouri J. Math. Sci. (2025) Vol. 37, No. 1, 67-78.

Cf. A014062, A014070, A086687.

[Binomial(n^n, n): n in [1..15]]; // Vincenzo Librandi, Apr 22 2011

Mathematica
```
Table[Binomial[n^n,n],{n,12}]
```

a(n) ~ n^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

A371471 a(n) = binomial(n^2,n) mod n^5.

Original entry on oeis.org

0, 6, 84, 796, 5, 3792, 7, 27768, 22608, 56440, 11, 83772, 13, 168448, 61065, 471536, 17, 445320, 19, 994080, 2258235, 4188272, 23, 6083208, 156275, 743912, 13548492, 3588928, 29, 16444800, 31, 28887008, 13685133, 2841992, 42053795, 49421088, 37, 24763840, 9171162

Offset: 1

Seiichi Manyama, Mar 24 2024

nonn

D. B. Fuks and Serge Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics, American Mathematical Society, 2007. Lecture 2. Arithmetical Properties of Binomial Coefficients, pages 27-44.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A014062, A081369.

A371471[n_] := Mod[Binomial[n^2, n], n^5];
Array[A371471, 50] (* Paolo Xausa, Jul 28 2025 *)

PARI
```
a(n) = binomial(n^2, n)%n^5;
```

If p is prime and p>=5, a(p) = p.

cp's OEIS Frontend

A383120 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k).

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Mathematica

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Formula

A055789 a(n) = binomial(n, round(sqrt(n))).

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Magma

Maple

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Sage

Formula

A066789 a(n) = binomial(n^3, n^2).

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Mathematica

PARI

Extensions

A107447 a(n) = binomial(n!, n^2).

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Magma

Mathematica

SageMath

Formula

A177424 Exponent of the highest power of 2 dividing binomial(n^2,n).

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Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A178842 a(n) = binomial((n-1)^2, n).

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GAP

Magma

Mathematica

PARI

Sage

A184357 a(n) = Sum_{k=0..n} C(n^2-k^2, n-k)*C(k^2, k).

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