A005810 -id:A005810 - cp's OEIS frontend

A364510 a(n) = binomial(4*n, n)^2.

1, 16, 784, 48400, 3312400, 240374016, 18116083216, 1401950721600, 110634634890000, 8862957169158400, 718528370729238784, 58818762721626513424, 4853704694918904043024, 403242220875862752160000, 33694913171561404510440000, 2829611125043050701300998400

Offset: 0

Peter Bala, Jul 28 2023

Paolo Xausa, Table of n, a(n) for n = 0..500
Wikipedia, Dixon's identity

Cf. A002894, A005810, A364506, A364511, A364512.

Row 2 of A364509.

Maple
```
seq( binomial(4*n,n)^2, n = 0..15);
```

A364510[n_]:=Binomial[4n,n]^2;Array[A364510,15,0] (* Paolo Xausa, Oct 05 2023 *)

a(n) = Sum_{i = -n..n} (-1)^i * binomial(2*n, n+i)^2 * binomial(4*n, 2*n+i).
Compare with Dixon's identity: Sum_{i = -n..n} (-1)^i * binomial(2*n, n+i)^3 = (3*n)!/n!^3.
a(n) = A005810(n)^2.
P-recursive: a(n) = 16 * ( (4*n - 1)*(4*n - 2)*(4*n - 3)/(3*n*(3*n - 1)*(3*n - 2)) )^2 * a(n-1) with a(0) = 1.
a(n) ~ c^n * 2/(3*Pi*n), where c = (2^16)/(3^6).
a(n) = [x^n] G(x)^(16*n), where the power series G(x) = 1 + x + 9*x^2 + 225*x^3 + 7525*x^4 + 295228*x^5 + 12787152*x^6 + 592477457*x^7 + 28827755219*x^8 + ... appears to have integer coefficients.
exp( Sum_{n > = 1} a(n)*x^n/n ) = F(x)^16, where the power series F(x) =  1 + x + 25*x^2 + 1033*x^3 + 53077*x^4 + 3081944*x^5 + 193543624*x^6 + 12835533333*x^7 + 886092805699*x^8 + ... appears to have integer coefficients.
The supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3^r)) hold for all primes p >= 5 and all positive integers n and r.
a(n) = [x^(3*n)] ( (1 - x)^(6*n)*Legendre_P(2*n, (1 + x)/(1 - x)) ). - Peter Bala, Aug 14 2023

A119304 Triangle read by rows: T(n,k) = binomial(4n-k,n-k), 0 <= k <= n.

Original entry on oeis.org

1, 4, 1, 28, 7, 1, 220, 55, 10, 1, 1820, 455, 91, 13, 1, 15504, 3876, 816, 136, 16, 1, 134596, 33649, 7315, 1330, 190, 19, 1, 1184040, 296010, 65780, 12650, 2024, 253, 22, 1, 10518300, 2629575, 593775, 118755, 20475, 2925, 325, 25, 1, 94143280, 23535820

Offset: 0

Paul Barry, May 13 2006

			Triangle begins
       1;
       4,     1;
      28,     7,    1;
     220,    55,   10,    1;
    1820,   455,   91,   13,   1;
   15504,  3876,  816,  136,  16,  1;
  134596, 33649, 7315, 1330, 190, 19, 1;

Indranil Ghosh, Rows 0..125, flattened

Rows sums are A052203. First column is A005810. Inverse of A119305.

Cf. A092392, A119301.

Flatten[Table[Binomial[4n-k,n-k],{n,0,9},{k,0,n}]] (* Indranil Ghosh, Feb 26 2017 *)

tabl(nn) = {for (n=0,nn,for (k=0,n,print1(binomial(4*n-k,n-k),", ");); print(););} \\ Indranil Ghosh, Feb 26 2017

from sympy import binomial
i=0
for n in range(12):
    for k in range(n+1):
        print(str(i)+" "+str(binomial(4*n-k,n-k)))
        i+=1 # Indranil Ghosh, Feb 26 2017

Riordan array (1/(1-4f(x)),f(x)) where f(x)(1-f(x))^3 = x.
From Peter Bala, Jun 04 2024: (Start)
'Horizontal' recurrence equation: T(n, 0) = binomial(4*n,n) and for k >= 1, T(n, k) = Sum_{i = 1..n+1-k} i*(i+1)/2 * T(n-1, k-2+i).
T(n, k) = Sum_{j = 0..n} binomial(n+j-1, j)*binomial(3*n-k-j, 2*n). (End)

A169959 a(n) = binomial(10*n, n).

Original entry on oeis.org

1, 10, 190, 4060, 91390, 2118760, 50063860, 1198774720, 28987537150, 706252528630, 17310309456440, 426342151127100, 10542859559688820, 261594860525768000, 6509613950241656640, 162392216278033616560, 4059949873964357469950, 101696990867999141755140

Offset: 0

N. J. A. Sloane, Aug 07 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..115

binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).

[Binomial(10*n, n): n in [0..50]]; // Vincenzo Librandi, Apr 21 2011

A169959[n_] := Binomial[10*n, n]; Array[A169959, 20, 0] (* Paolo Xausa, Aug 20 2025 *)

a(n) = C(10*n-1, n-1)*C(100*n^2, 2)/(3*n*C(10*n+1, 3)), n > 0. - Gary Detlefs, Jan 02 2014
From Peter Bala, Feb 21 2022: (Start)
The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 9*A(x))^9 +  (10^10)*x*A(x)^10 = 0.
Sum_{n >= 1} a(n)*( x*(9*x + 10)^9/(10^10*(1 + x)^10) )^n = x. (End)

A264773 Triangle T(n,k) = binomial(4n - 3k, 3n - 2k), 0 <= k <= n.

Original entry on oeis.org

1, 4, 1, 28, 5, 1, 220, 36, 6, 1, 1820, 286, 45, 7, 1, 15504, 2380, 364, 55, 8, 1, 134596, 20349, 3060, 455, 66, 9, 1, 1184040, 177100, 26334, 3876, 560, 78, 10, 1, 10518300, 1560780, 230230, 33649, 4845, 680, 91, 11, 1, 94143280, 13884156, 2035800, 296010, 42504, 5985, 816, 105, 12, 1

Offset: 0

Peter Bala, Nov 30 2015

Riordan array (f(x),x*g(x)), where g(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + ... is the o.g.f. for A002293 and f(x) = g(x)/(4 - 3*g(x)) = 1 + 4*x + 28*x^2 + 220*x^3 + 1820*x^4 + ... is the o.g.f. for A005810.

More generally, if (R(n,k))n,k>=0 is a proper Riordan array and m is a nonnegative integer and a > b are integers then the array with (n,k)-th element R((m + 1)*n - a*k, m*n - b*k) is also a Riordan array (not necessarily proper). Here we take R as Pascal's triangle and m = a = 3 and b = 2. See A092392, A264772, A264774 and A113139 for further examples.

			Triangle begins
  n\k |       0      1     2    3   4   5   6   7
------+-----------------------------------------------
   0  |       1
   1  |       4      1
   2  |      28      5     1
   3  |     220     36     6    1
   4  |    1820    286    45    7   1
   5  |   15504   2380   364   55   8   1
   6  |  134596  20349  3060  455  66   9   1
   7  | 1184040 177100 26334 3876 560  78  10   1
...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).
Peter Bala, A 4-parameter family of embedded Riordan arrays
E. Lebensztayn, On the asymptotic enumeration of accessible automata, Discrete Mathematics and Theoretical Computer Science, Vol. 12, No. 3, 2010, 75-80, Section 2.
R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics CreateSpace Independent Publishing Platform 2006, Section 5.6, ISBN-13: 978-1502925244.

A005810 (column 0), A052203 (column 1), A257633 (column 2), A224274 (column 3), A004331 (column 4). Cf. A002293, A007318, A092392 (C(2n-k,n)), A119301 (C(3n-k,n-k)), A264772, A264774.

/* As triangle: */ [[Binomial(4*n-3*k, 3*n-2*k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Dec 02 2015

A264773:= proc(n,k) binomial(4*n - 3*k, 3*n - 2*k); end proc:
seq(seq(A264773(n,k), k = 0..n), n = 0..10);

A264773[n_,k_] := Binomial[4*n - 3*k, n - k];
Table[A264773[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 06 2024 *)

T(n,k) = binomial(4*n - 3*k, n - k).

O.g.f.: f(x)/(1 - t*x*g(x)), where f(x) = Sum_{n >= 0} binomial(4*n,n)*x^n and g(x) = Sum_{n >= 0} 1/(3*n + 1)*binomial(4*n,n)*x^n.

A357955 a(n) = 3binomial(4n,n) - 20binomial(3n,n) + 54binomial(2n,n).

Original entry on oeis.org

37, 60, 108, 60, -660, 60, 82404, 1411848, 17540460, 191318820, 1952058108, 19175376324, 184118073828, 1743153802320, 16359157606200, 152693295412560, 1420516291306860, 13190159377278324, 122358232382484420, 1134645084249344400, 10522118980232969340

Offset: 0

Peter Bala, Oct 22 2022

Conjectures:
1) a(p) == a(1) (mod p^7) for all primes p >= 3 except p = 7.
2) a(p^r) == a(p^(r-1)) ( mod p^(3*r+5) ) for r >= 2 and all primes p >= 3.
These are stronger supercongruences than those satisfied separately by the sequences {binomial(2*n,n)} = A000984, {binomial(3*n,n)} = A005809 and {binomial(4*n,n)} = A005810.
Conjecture 1) was proved by Aidagulov and Alekseyev; see the remarks following Corollary 2. - Peter Bala, Oct 29 2022

			Examples of supercongruences:
a(11) - a(1) = 19175376324 - 60 = (2^3)*3*(11^7)*41 == 0 (mod 11^7).
a(5^2) - a(5) = 726506045044361132812560 - 60 = (2^2)*3*(5^11)*41*30241552444123 == 0 (mod 5^11).

Paolo Xausa, Table of n, a(n) for n = 0..1000
R. R. Aidagulov and M. A. Alekseyev, On p-adic approximation of sums of binomial coefficients, Journal of Mathematical Sciences 233:5 (2018), 626-634; arXiv:1602.02632 [math.NT], 2018.
C. Helou and G. Terjanian, On Wolstenholme’s theorem and its converse, J. Number Theory 128 (2008), 475-499.
Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv preprint arXiv:1111.3057 [math.NT], 2011.

Cf. A000984, A005809, A005810, A268590, A357509, A357567, A357568, A357569.

seq(3*binomial(4*n,n) - 20*binomial(3*n,n) + 54*binomial(2*n,n), n = 0..20);

A357955[n_] := 3*Binomial[4*n, n] - 20*Binomial[3*n, n] + 54*Binomial[2*n, n];
Array[A357955, 25, 0] (* Paolo Xausa, Jul 17 2024 *)

from math import comb
def A357955(n): return 54*comb(m:=n<<1,n)+3*comb(m<<1,n)-20*comb(m+n,n) # Chai Wah Wu, Oct 24 2022

a(n) = 3*A005810(n) - 20*A005809(n) + 54*A000984(n).
a(k*p^r) == a(k*p^(r-1)) ( mod p^(3*r) ) for positive integers k and r and for all primes p >= 5 (see Meštrović, Section 6, equation 39).
a(p) == a(1) (mod p^6) for all primes p >= 7 (apply Helou and Terjanian, Section 3, Proposition 2).
From Stefano Spezia, Jul 17 2024: (Start)
G.f.: 54/sqrt(1-4*x) - 40*cos(arccos(1-27*x/2)/6)/sqrt(4-27*x) + 3*hypergeom([1/4, 1/2, 3/4], [1/3, 2/3], 4^4*x/3^3).
E.g.f.: 54*exp(2*x)*BesselI(0, 2*x) - 20*hypergeom([1/3, 2/3], [1/2, 1], 27*x/4) + 3*hypergeom([1/4, 1/2, 3/4], [1/3, 2/3, 1], 4^4*x/3^3).
a(n) ~ exp(2*n*arctanh(229/283))*sqrt(6/(n*Pi)). (End)

A371814 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*n-k-1,n-k).

Original entry on oeis.org

1, 2, 16, 128, 1068, 9142, 79612, 701864, 6244892, 55962920, 504375396, 4567003520, 41513817444, 378596616452, 3462411408136, 31742042431048, 291616814436124, 2684123914512280, 24746511514749280, 228491677484832896, 2112549277665243328

Offset: 0

Seiichi Manyama, Apr 06 2024

nonn

Cf. A072547, A371813.
Cf. A005810, A262977.
Cf. A066381, A262977, A385498, A386701.

a(n) = sum(k=0, n, (-1)^k*binomial(4*n-k-1, n-k));

a(n) = [x^n] 1/((1+x) * (1-x)^(3*n)).
a(n) = binomial(4*n-1, n)*hypergeom([1, -n], [1-4*n], -1). - Stefano Spezia, Apr 07 2024
From Vaclav Kotesovec, Apr 07 2024: (Start)
Recurrence: 24*n*(3*n - 2)*(3*n - 1)*(415*n^3 - 1898*n^2 + 2871*n - 1436)*a(n) = (838715*n^6 - 5099533*n^5 + 12225995*n^4 - 14652035*n^3 + 9157250*n^2 - 2799192*n + 322560)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(415*n^3 - 653*n^2 + 320*n - 48)*a(n-2).
a(n) ~ 2^(8*n + 1/2) / (5 * sqrt(Pi*n) * 3^(3*n - 1/2)). (End)
a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n,k). - Seiichi Manyama, Jul 30 2025
G.f.: g/((-1+2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 13 2025
a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025
G.f.: 1/(1 - x*g^3*(-4+6*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

A378804 a(n) = n * 2^n * binomial(4*n, n).

Original entry on oeis.org

0, 8, 224, 5280, 116480, 2480640, 51684864, 1060899840, 21541478400, 433812234240, 8680043806720, 172774871965696, 3424347806171136, 67626404043161600, 1331466198928588800, 26145958720005734400, 512257621575157678080, 10016204637370583089152, 195501127311163895316480

Offset: 0

Amiram Eldar, Dec 07 2024

Amiram Eldar, Table of n, a(n) for n = 0..500
Jonathan M. Borwein and Roland Girgensohn, Evaluations of binomial series, aequationes mathematicae, Vol. 70, No. 1 (2005), pp. 25-36.

Cf. A005810, A036289, A378802.

a[n_] := n * 2^n * Binomial[4*n, n]; Array[a, 20, 0]

PARI
```
a(n) = n * 2^n * binomial(4*n, n);
```

a(n) = A036289(n) * A005810(n).
a(n) = 2^n * A378802(n).
a(n) == 0 (mod 8).
Sum_{n>=1} (-1)^n/a(n) = (log(2) - 6*log(3))/7 + Sum_{r: 2*r^3 + 12*r + 13 = 0} log(r+2)/(r+3) = -0.120716907732393305... (Borwein and Girgensohn, 2005, p. 32, eq. (43)).

A386918 a(n) = 2^n * binomial(4*n,n).

Original entry on oeis.org

1, 8, 112, 1760, 29120, 496128, 8614144, 151557120, 2692684800, 48201359360, 868004380672, 15706806542336, 285362317180928, 5202031080243200, 95104728494899200, 1743063914667048960, 32016101348447354880, 589188508080622534656, 10861173739509105295360

Offset: 0

Seiichi Manyama, Aug 08 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..350

Cf. A066381, A386919, A386920.

Cf. A005810, A059304, A228484, A378804.

[2^n * Binomial(4*n,n): n in [0..26]]; // Vincenzo Librandi, Aug 11 2025

Table[2^n*Binomial[4*n,n],{n,0,30}] (* Vincenzo Librandi, Aug 11 2025 *)

PARI
```
a(n) = 2^n*binomial(4*n, n);
```

a(n) = Sum_{k=0..n} binomial(4*n,k) * binomial(4*n-k,n-k).
a(n) = [x^n] (1+x)^(4*n)/(1-x)^(3*n+1).
a(n) = [x^n] 1/(1-2*x)^(3*n+1).
a(n) = [x^n] (1+2*x)^(4*n).

A387085 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*n+1,k).

Original entry on oeis.org

1, 0, 4, 8, 36, 120, 456, 1680, 6340, 23960, 91224, 348656, 1337896, 5149872, 19877904, 76907808, 298176516, 1158168792, 4505865144, 17555689008, 68490100536, 267518448912, 1046041377264, 4094231982048, 16039426479336, 62887835652720, 246761907761776, 968943740083040

Offset: 0

Seiichi Manyama, Aug 16 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000302, A000984, A001700, A026641, A141223, A377011, A386957.
Cf. A226733, A226705, A226751.
Cf. A005810, A079589, A183160.

[&+[(-3)^(n-k) * Binomial(2*n+1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[(-3)^(n-k)*Binomial[2*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)

a(n) = sum(k=0, n, (-3)^(n-k)*binomial(2*n+1, k));

a(n) = [x^n] (1+x)^(2*n+1)/(1+3*x).
a(n) = [x^n] 1/((1-x)^(n+1) * (1+2*x)).
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} (-2)^k * binomial(2*n-k,n-k).
G.f.: 1/( 4*x - 1 + 2*sqrt(1 - 4*x) ).
G.f.: 1/(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: g^2/((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: B(x)^2/(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.
D-finite with recurrence 3*n*a(n) +2*(-4*n+3)*a(n-1) +8*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Aug 19 2025

A026020 a(n) = binomial(4n, n) - binomial(4n, n - 3).

Original entry on oeis.org

1, 4, 28, 219, 1804, 15314, 132572, 1163565, 10316924, 92195488, 829016968, 7492106505, 67991427828, 619193535380, 5655829748520, 51794730347745, 475390078267356, 4371917301657488, 40276635724273936, 371630891401943020, 3433826368544377520, 31768260456301092090

Offset: 0

Clark Kimberling

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

a(n) = T(4n, n), where T is the array defined in A026009.
Bisections are A026012 and A026016.
Cf. A004333, A005810.

[Binomial(4*n, n) - Binomial(4*n, n-3): n in [0..20]]; // G. C. Greubel, Mar 22 2021

A026020:= n-> binomial(4*n,n) - binomial(4*n,n-3); seq(A026020(n), n=0..20); # G. C. Greubel, Mar 22 2021

Table[Binomial[4n, n] - Binomial[4n, n - 3], {n, 0, 19}] (* Alonso del Arte, Jun 06 2019 *)

a(n) = binomial(4*n, n) - binomial(4*n, n-3) \\ Felix Fröhlich, Jun 06 2019

[binomial(4*n, n) - binomial(4*n, n-3) for n in (0..20)] # G. C. Greubel, Mar 22 2021

G.f.: (g - 2)*(1 - g + g^2)*g/(3*g - 4) where g = 1 + x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
a(n) = A005810(n) - A004333(n) for n > 2 - Felix Fröhlich, Jun 06 2019
a(n) ~ 13 * 2^(8*n+3/2) / (3^(3*n+7/2) * sqrt(Pi*n)). - Amiram Eldar, Sep 06 2025

cp's OEIS Frontend

A364510 a(n) = binomial(4*n, n)^2.

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A119304 Triangle read by rows: T(n,k) = binomial(4n-k,n-k), 0 <= k <= n.

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A169959 a(n) = binomial(10*n, n).

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A264773 Triangle T(n,k) = binomial(4*n - 3*k, 3*n - 2*k), 0 <= k <= n.

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A357955 a(n) = 3*binomial(4*n,n) - 20*binomial(3*n,n) + 54*binomial(2*n,n).

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A371814 a(n) = Sum_{k=0..n} (-1)^k * binomial(4*n-k-1,n-k).

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A378804 a(n) = n * 2^n * binomial(4*n, n).

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A386918 a(n) = 2^n * binomial(4*n,n).

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A264773 Triangle T(n,k) = binomial(4n - 3k, 3n - 2k), 0 <= k <= n.

A357955 a(n) = 3binomial(4n,n) - 20binomial(3n,n) + 54binomial(2n,n).