A002894 -id:A002894 - cp's OEIS frontend

A008978 a(n) = (5*n)!/(n!)^5.

1, 120, 113400, 168168000, 305540235000, 623360743125120, 1370874167589326400, 3177459078523411968000, 7656714453153197981835000, 19010638202652030712978200000, 48334775757901219912115629238400, 125285878026462826569986857692288000

Offset: 0

N. J. A. Sloane

Number of paths of length 5n in Z^5 from (0,0,0,0,0) to (n,n,n,n,n).

Vincenzo Librandi, Table of n, a(n) for n = 0..100
V. Batyrev, Review of "Mirror Symmetry and Algebraic Geometry", by D. A. Cox and S. Katz, Bull. Amer. Math. Soc., 37 (No. 4, 2000), 473-476.
R. M. Dickau, 5-D shortest path diagrams
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. A000984, A006480, A008977, A002894, A002897, A186420, A188662, A001460.

Row 5 of A187783.

[Factorial(5*n)/Factorial(n)^5: n in [0..10]]; // Vincenzo Librandi, Mar 08 2014

Table[(5 n)!/(n)!^5, {n, 0, 20}] (* Vincenzo Librandi, Mar 08 2014 *)

a(n) = (5*n)!/(n!)^5; \\ Michel Marcus, Mar 08 2014

a(n) ~ 5^(5*n+1/2) / (4 * Pi^2 * n^2). - Vaclav Kotesovec, Mar 07 2014
From Peter Bala, Jul 12 2016: (Start)
a(n) = binomial(2*n,n)*binomial(3*n,n)*binomial(4*n,n)*binomial(5*n,n) = ( [x^n](1 + x)^(2*n) ) * ( [x^n](1 + x)^(3*n) ) * ( [x^n](1 + x)^(4*n) ) * ( [x^n](1 + x)^(5*n) ) = [x^n]( F(x)^(120*n) ), where F(x) = 1 + x + 353*x^2 + 318986*x^3 + 408941594*x^4 + 633438203535*x^5 + 1105336091531052*x^6 + ... appears to have integer coefficients. For similar results see A000897, A002894, A002897, A006480, A008977, A186420 and A188662. (End)
From Peter Bala, Jul 17 2016: (Start)
a(n) = Sum_{k = 0..4*n} (-1)^k*binomial(5*n,n + k)*binomial(n + k,k)^5.
a(n) = Sum_{k = 0..5*n} (-1)^(n+k)*binomial(5*n,k)*binomial(n + k,k)^5. (End)
From Ilya Gutkovskiy, Nov 23 2017: (Start)
O.g.f.: 4F3(1/5,2/5,3/5,4/5; 1,1,1; 3125*x).
E.g.f.: 4F4(1/5,2/5,3/5,4/5; 1,1,1,1; 3125*x). (End)
From Peter Bala, Feb 16 2020: (Start)
a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k - apply Mestrovic, equation 39, p. 12.
a(n) = [(x*y*z*u)^n] (1 + x + y + z + u )^(5*n). (End)
a(n) = 120*A322252(n). - R. J. Mathar, Jun 21 2023
a(n) = a(n-1)*5*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4)/n^4. - Neven Sajko, Jul 21 2023

A060150 a(0) = 1; for n > 0, binomial(2n-1, n-1)^2.

Original entry on oeis.org

1, 1, 9, 100, 1225, 15876, 213444, 2944656, 41409225, 590976100, 8533694884, 124408576656, 1828114918084, 27043120090000, 402335398890000, 6015361252737600, 90324408810638025, 1361429497505672100, 20589520178326522500, 312321918272897610000

Offset: 0

N. J. A. Sloane, Apr 10 2001

Number of square lattice walks that start at (0,0) and end at (1,0) after 2n-1 steps, free to pass through (1,0) at intermediate steps. - Steven Finch, Dec 20 2001
Number of paths of length n connecting two neighboring nodes in optimal chordal graph of degree 4, G(2*d(G)^2+2*d(G)+1,2d(G)+1), of diameter d(G). - B. Dubalski (dubalski(AT)atr.bydgoszcz.pl), Feb 05 2002
a(n) is the number of ways to place n red balls and n blue balls into n distinguishable boxes with no restrictions on the number of balls put in a box. - Geoffrey Critzer, Jul 08 2013
The number of square lattice walks of n steps that start at the origin and end at (k,0) is zero if n-k is odd and [binomial(n,(n-k)/2)]^2 if n-k is even. - R. J. Mathar, Sep 28 2020

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 1994 Addison-Wesley company, Inc.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (5.1.29.2)
K. A. Ross and C. R. B. Wright, Discrete Mathematics, 1992 Prentice Hall Inc.

Harry J. Smith, Table of n, a(n) for n = 0..200
R. Bacher, Meander algebras

Cf. A002894, A135389, A337900, A337901, A337902.

seq(coeff(series(EllipticK(4*sqrt(x))/(2*Pi) + 3/4, x=0, n+1), x, n), n=0..30);  # Mark van Hoeij, Apr 30 2013

Table[Binomial[2n-1,n]^2,{n,0,19}] (* Geoffrey Critzer, Jul 08 2013 *)

PARI
```
a(n)=if(n<2, 1, binomial(2*n-1,n-1)^2)
```

for (n=0, 200, if (n==0, a=1, a=binomial(2*n - 1, n - 1)^2); write("b060150.txt", n, " ", a)) \\ Harry J. Smith, Jul 02 2009

a(n) = A088218(n)^2.
a(n) = A002894(n)/4 for n>0.
G.f.: 1 + (1/AGM(1, sqrt(1-16*x))-1)/4. - Michael Somos, Dec 12 2002
G.f. = 1 + (K(16x)-1)/4 = 1 + Sum_{k>0} q^k/(1+q^(2k)) where K(16x) is the complete Elliptic integral of the first kind at 16x=k^2 and q is the nome. - Michael Somos, May 09 2005
G.f.: 1 + x*3F2((1, 3/2, 3/2); (2, 2))(16*x). - Olivier Gérard, Feb 16 2011
E.g.f.: Sum_{n>0} a(n)*x^(2n-1)/(2n-1)! = BesselI(0, 2x)*BesselI(1, 2x) . - Michael Somos, Jun 22 2005
D-finite with recurrence n^2*a(n) -4*(2*n-1)^2*a(n-1)=0. - R. J. Mathar, Jul 26 2014
From Seiichi Manyama, Oct 19 2016: (Start)
Let the number of multisets of length k on n symbols be denoted by ((n, k)) = binomial(n+k-1, k).
a(n) = (Sum_{0 <= k <= n} binomial(n, k)^2 * ((2*n, n - k)))/3 for n > 0. (End)
a(n) ~ 4^(2*n-1)/(Pi*n). - Ilya Gutkovskiy, Oct 19 2016
For n >= 1, a(n) = 1/n * Sum_{k = 0..n-1} (n + 2*k)*binomial(n+k-1, k)^2 = ( 1/(4*n) * Sum_{k = 0..n} (n + 2*k)*binomial(-n+k-1, k)^2 )^2. - Peter Bala, Nov 02 2024

A091401 Numbers n such that genus of group Gamma_0(n) is zero.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25

Offset: 1

N. J. A. Sloane, Mar 02 2004

Equivalently, numbers n such that genus of modular curve X_0(n) is zero.

G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.

Miranda C. N. Cheng, John F. R. Duncan and Jeffrey A Harvey, Umbral moonshine and the Niemeier lattices, Research in the Mathematical Sciences, 2014, 1:3; See Eq. (22). - _N. J. A. Sloane_, Jun 19 2014
K. Harada, "Moonshine" of Finite Groups, European Math. Soc., 2010, p. 15.
Yang-Hui He, John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.
K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, vol. 102, American Mathematical Society, Providence, RI, 2004. See p. 110.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.

Cf. A001617, A001615, A000089, A000086, A001616, A091403.
The table below is a consequence of Theorem 7.3 in Maier's paper.
N        EntryID        K       alpha
1
      A127776        4096    1
      A276018        729     1
      A002894        256     1
      A276019        125     4
      A093388        72      1
      A276021        49      9
      A081085        32      1
      A006077        27      1
     A276020        20      2
     A276022        12      1
     A276177        13      36
     A276178        8       1
     A276179        6       1
     A276180        5       4

Flatten@ Position[#, 0] &@ Table[If[n < 1, 0, 1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors@ n}] - Count[(#^2 - # + 1)/n & /@ Range@ n, ?IntegerQ]/3 - Count[(#^2 + 1)/n & /@ Range@ n, ?IntegerQ]/4], {n, 120}] (* Michael De Vlieger, Dec 05 2016, after Michael Somos at A001617 *)

Numbers n such that A001617(n) = 0.

A000894 a(n) = (2n)!(2n+1)! /((n+1)! n!^3).

Original entry on oeis.org

1, 6, 60, 700, 8820, 116424, 1585584, 22084920, 312869700, 4491418360, 65166397296, 953799087696, 14062422446800, 208618354980000, 3111393751416000, 46619049708716400, 701342468412012900

Offset: 0

N. J. A. Sloane

This sequence is one half of the odd part of the bisection of A241530. The even part is given in A002894. - Wolfdieter Lang, Sep 06 2016

			G.f. = 1 + 6*x + 60*x^2 + 700*x^3 + 8820*x^4 + 116424*x^5 + ...

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 96.

Vincenzo Librandi, Table of n, a(n) for n = 0..180
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See p. 36.
Pedro J. Miana and Natalia Romero, Moments of combinatorial and Catalan numbers, Journal of Number Theory, Volume 130, Issue 8, August 2010, Pages 1876-1887. See Omega1 Remark 3 p. 1882.
Yidong Sun and Fei Ma, Four transformations on the Catalan triangle, arXiv preprint arXiv:1305.2017 [math.CO], 2013 (see Omega_1).
Yidong Sun and Fei Ma, Some new binomial sums related to the Catalan triangle, Electronic Journal of Combinatorics 21(1) (2014), #P1.33.

First differences of A082578.

Cf. A000108, A000142, A000217, A000891, A000984, A001700, A002463, A002894, A103371, A132813, A241530, A248845.

a000894 n = a132813 (2 * n) n  -- Reinhard Zumkeller, Apr 04 2014

[Factorial(2*n)*Factorial(2*n+1) /(Factorial(n+1)* Factorial(n)^3): n in [0..20]]; // Vincenzo Librandi, Oct 25 2011

A000894:= func< n | Binomial(2*n+2,2)*Catalan(n)^2 >;
[A000894(n): n in [0..40]]; // G. C. Greubel, Mar 12 2025

seq(binomial(2*n+1,n)*binomial(2*n,n), n=0..16); # Zerinvary Lajos, Jan 23 2007

a[ n_] := Binomial[2 n + 1, n] Binomial[2 n, n]; (* Michael Somos, May 28 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticK[ 16 x] - EllipticE[ 16 x]) / (4 x Pi), {x, 0, n}]; (* Michael Somos, May 28 2014 *)
Table[(2 n)!*(2 n + 1)!/((n + 1)!*n!^3), {n, 0, 16}] (* Michael De Vlieger, Sep 06 2016 *)

{a(n) =  binomial( 2*n + 1, n) * binomial( 2*n, n)}; /* Michael Somos, May 28 2014 */

def A000894(n): return binomial(2*n+2,2)*catalan_number(n)^2
print([A000894(n) for n in range(41)]) # G. C. Greubel, Mar 12 2025

From Zerinvary Lajos, Jan 23 2007: (Start)
a(n) = C(2*n+1,n)*C(2*n,n) = A001700(n)*A000984(n).
a(n) = A000984(n)*A000984(n+1)/2, n>=0. (End)
G.f.: (EllipticK(4*sqrt(x)) - EllipticE(4*sqrt(x)))/(4*Pi*x). - Mark van Hoeij, Oct 24 2011
n*(n+1)*a(n) = 4*(2*n-1)*(2*n+1)*a(n-1). - R. J. Mathar, Sep 08 2013
a(n) = A103371(2*n,n) = A132813(2*n,n). - Reinhard Zumkeller, Apr 04 2014
0 = a(n)*(+65536*a(n+2) - 23040*a(n+3) + 1400*a(n+4)) + a(n+1)*(-1536*a(n+2) + 1184*a(n+3) - 90*a(n+4)) + a(n+2)*(-24*a(n+2) - 6*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, May 28 2014
0 = a(n+1)^3 * (+256*a(n) - 6*a(n+1) + a(n+2)) + a(n) * a(n+1) * a(n+
2) * (-768*a(n) - 20*a(n+1) - 3*a(n+2)) + 90*a(n)^2*a(n+2)^2 for all n in Z. - Michael Somos, Sep 17 2014
a(n) = (n+1) * A000891(n) = A248045(n+1) / A000142(n). - Reinhard Zumkeller, Sep 30 2014
a(n) = A241530(2n+1)/2, n >= 0. - Wolfdieter Lang, Sep 06 2016
a(n) ~ 2^(4*n+1)/(Pi*n). - Ilya Gutkovskiy, Sep 06 2016
a(n) = A000217(n+1)*A000108(n)*A000108(n+1) = A000217(2*n+1)*A000108(n)^2. - G. C. Greubel, Mar 12 2025

A000888 a(n) = (2n)!^2 / ((n+1)!n!^3).

Original entry on oeis.org

1, 2, 12, 100, 980, 10584, 121968, 1472328, 18404100, 236390440, 3103161776, 41469525552, 562496897872, 7726605740000, 107289439704000, 1503840313184400, 21252802073091300, 302539888334593800, 4334635827016110000, 62464383654579522000, 904841214653480504400

Offset: 0

N. J. A. Sloane

a(n) is the number of walks of 2n unit steps North, East, South, or West, starting at the origin, bounded above by y=x, below by y=-x and terminating on the ray y = x >= 0. Example: a(1) counts EN, EW; a(2) counts ESNN, ESNW, ENSN, ENSW, ENEN, ENEW, EENN, EENW, EEWN, EEWW, EWEN, EWEW. - David Callan, Oct 11 2005
Bijective proof: given such an NESW walk, construct a pair (P_1, P_2) of lattice paths of upsteps U=(1,1) and downsteps D=(1,-1) as follows. To get P_1, replace each E and S with U and each W and N with D. To get P_2, replace each N and E with U and each S and W with D. For example, EENSNW -> (UUDUDD, UUUDUD). This mapping is 1-to-1 and its range is the Cartesian product of the set of Dyck n-paths and the set of nonnegative paths of length 2n. The Dyck paths are counted by the Catalan number C_n (A000108) and the nonnegative paths are counted (see for example the Callan link) by the central binomial coefficient binomial(2n,n) (A000984). So this is a bijection from these NESW walks to a set of size C_n*binomial(2n,n) = a(n). - David Callan, Sep 18 2007
If A is a random matrix in USp(4) (4 X 4 complex matrices that are unitary and symplectic), then a(n) = E[(tr(A^3))^{2n}]. - Andrew V. Sutherland, Apr 01 2008
Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of 2 n steps taken from {(-1,-1), (-1,1), (1,-1), (1,1)}. - Manuel Kauers, Nov 18 2008
a(n) is equal to the n-th moment of the following positive function defined on x in (0,16), in Maple notation: (EllipticK(sqrt(1-x/16)) - EllipticE(sqrt(1-x/16)))/(Pi^2*sqrt(x)). This is the solution of the Hausdorff moment problem and thus it is unique. - Karol A. Penson, Feb 11 2011
The partial sums of a(n)/A013709(n) absolutely converge to 1/Pi. - Ralf Steiner, Jan 21 2016

			G.f.: 1 + 2*x + 12*x^2 + 100*x^3 + 980*x^4 + 10584*x^5 + 121968*x^6 + ...

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 93.
T. M. MacRobert, Functions of a Complex Variable, 4th ed., Macmillan & Co., London, 1958, p. 177.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Marco S. Bianchi, Protected and uniformly transcendental, arXiv:2306.06239 [hep-th], 2023.
M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
David Callan, Bijections for the identity 4^n = ... .
Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
Helmut Prodinger, Two New Identities Involving the Catalan Numbers: A classical approach, arXiv:1911.07604 [math.CO], 2019.
Ralf Steiner, Beispiele zur modifizierten Wallis-Lambert-Reihe, 2016.

Cf. A000108, A002894, A088538, A089835, A125558.

[(Factorial(2*n))^2/(Factorial(n))^4/(n+1): n in [0..20]]; // Vincenzo Librandi, Aug 15 2011

[seq(binomial(2*n,n)^2/(n+1),n=0..17)]; # Zerinvary Lajos, May 27 2006

f[n_] := Binomial[2 n, n]^2/(n + 1); Array[f, 18, 0]  (* Robert G. Wilson v *)
a[ n_] := SeriesCoefficient[ (1/8) (EllipticE[ 16 x] - (1 - 16 x) EllipticK[ 16 x]) / (Pi/2), {x, 0, n + 1}]; (* Michael Somos, Jan 23 2012 *)

{a(n) = if( n<0, 0, (2*n)!^2 / n!^4 / (n+1))}; /* Michael Somos, Sep 11 2005 */

G.f.: 1/4*((16*x-1)*EllipticK(4*x^(1/2)) + EllipticE(4*x^(1/2)))/x/Pi. - Vladeta Jovovic, Oct 12 2003
Given G.f. A(x), y = x*A(x) satisfies y = y'' * (1 - 16*x) * x/4. - Michael Somos, Sep 11 2005
a(n) = binomial(2*n,n)^2/(n+1). - Zerinvary Lajos, May 27 2006
G.f.: 2F1(1/2,1/2;2;16*x). - Paul Barry, Sep 03 2008
a(n) = 2*A125558(n) (n >= 1). - Olivier Gérard, Feb 16 2011
A002894(n) = (n+1) * a(n). A001246(n) = a(n) / (n+1). A089835(n) = n! * a(n). - Michael Somos, May 12 2012
G.f.: 1 + 4*x/(G(0)-4*x) where G(k) = 4*x*(2*k+1)^2 + (k+1)*(k+2) - 4*x*(k+1)*(k+2)*(2*k+3)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 30 2012
D-finite with recurrence: (n+1)*(n+2)*a(n+1) = 4*(2*n+1)^2*a(n). - Vaclav Kotesovec, Sep 11 2012
a(n) = C(n)*binomial(2*n,n) = Sum_{k=0..2*n} binomial(2*n,k)*C(k)*C(2*n-k) where C(k) are Catalan numbers (A000108), see Prodinger. - Michel Marcus, Nov 19 2019
Sum_{n>=0} a(n)/16^n = 4/Pi (A088538). - Amiram Eldar, May 06 2023

A054474 Number of walks on square lattice that start and end at origin after 2n steps, not touching origin at intermediate stages.

Original entry on oeis.org

1, 4, 20, 176, 1876, 22064, 275568, 3584064, 47995476, 657037232, 9150655216, 129214858304, 1845409805168, 26606114089024, 386679996988736, 5658611409163008, 83302885723872852, 1232764004638179504, 18327520881735288432, 273595871825723062848

Offset: 0

Alessandro Zinani (alzinani(AT)tin.it), May 19 2000

1-dimensional and 3-dimensional analogs are A002420 and A049037.
Trajectories returning to the origin are prohibited, contrary to the situation in A094061.
The probability of returning to the origin for the first time after 2n steps is given by a(n)/4^(2*n). If A(x) is a generating function for this sequence, A(x/16) is a generating function for the sequence of probabilities. The sum of these probabilities for n > 0 is 1 unlike in dimensions > 2. - Shel Kaphan, Feb 13 2023

			a(5)=22064, i.e., there are 22064 different walks (on a square lattice) that start and end at the origin after 2*5=10 steps, avoiding the origin at intermediate steps.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.

Alois P. Heinz, Table of n, a(n) for n = 0..834
S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice [Broken link]
Steven R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice [Cached copy, with permission of the author]
Markus Kuba and Alois Panholzer, Lattice paths and the diagonal of the cube, arXiv:2411.03930 [math.CO], 2024. See p. 14.

Cf. A002894, A002420, A049037, A063887.

Column k=2 of A361397.

b:= proc(n) b(n):= binomial(2*n, n)^2 end:
a:= proc(n) option remember;
      b(n)-add(a(n-i)*b(i), i=1..n-1)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Dec 05 2023

m = 18; gf[x_] = 2 - Pi/(2*EllipticK[4*Sqrt[x]]); (List @@ Normal[ Series[ gf[x], {x, 0, m-1}]] /. x -> 1)[[1 ;; m+1]]*Table[4^k, {k, 0, m}] (* Jean-François Alcover, Jun 16 2011, after Vladeta Jovovic *)
CoefficientList[Series[2-Pi/(2*EllipticK[16*x]),{x,0,20}],x] (* Vaclav Kotesovec, Mar 10 2014 *)
CoefficientList[Series[2-ArithmeticGeometricMean[1,Sqrt[1-16x]],{x,0,20}],x] (* Thomas Dybdahl Ahle, Oct 30 2023 *)

a(n)=if(n<0,0,polcoeff(2-agm(1,sqrt(1-16*x+x*O(x^n))),n))

G.f.: 2 - AGM(1, (1-16*x)^(1/2)).
G.f.: 2 - 1/hypergeom([1/2,1/2],[1],16*x). - Joerg Arndt, Jun 16 2011
Let (in Maple notation) G(x):=4/Pi*EllipticK(4*t)-2/Pi*EllipticF(1/sqrt(2+4*t), 4*t)-2/Pi*EllipticF(1/sqrt(2-4*t), 4*t), then GenFunc(x):=2-1/G(x). - Sergey Perepechko, Sep 11 2004
G.f.: 2 - Pi/(2*EllipticK(4*sqrt(x))). - Vladeta Jovovic, Jun 23 2005
a(n) ~ Pi * 16^n / (n * log(n)^2) * (1 - (2*gamma + 8*log(2)) / log(n) + (3*gamma^2 + 24*log(2)*gamma + 48*log(2)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019
INVERTi transform of A002894. - R. J. Mathar, Sep 24 2020

A188662 Binomial coefficients: a(n) = binomial(3*n,n)^2.

Original entry on oeis.org

1, 9, 225, 7056, 245025, 9018009, 344622096, 13521038400, 540917591841, 21966328580625, 902702926350225, 37456461988358400, 1566697064677290000, 65973795093338597136, 2794203818390077646400, 118933541228935777741056, 5084343623375056062840609

Offset: 0

Emanuele Munarini, Apr 07 2011

Even-order terms in the diagonal of rational function 1/(1 - (x^2 + y^2 + z^2 - x*y - y*z - x*z)). - Gheorghe Coserea, Aug 09 2018

Seiichi Manyama, Table of n, a(n) for n = 0..604

Cf. A005809, A000897, A002894, A002897, A006480, A008977, A186420.

[Binomial(3*n,n)^2: n in [0..100]]; // Vincenzo Librandi, Apr 08 2011

Mathematica
```
Table[Binomial[3 n, n]^2, {n, 0, 16}]
```
Maxima
```
makelist(binomial(3*n,n)^2,n,0,16);
```

a(n) = binomial(3*n,n)^2; \\ Michel Marcus, Nov 01 2016

from math import comb
def A188662(n): return comb(3*n,n)**2 # Chai Wah Wu, Mar 15 2023

a(n) = A005809(n)^2.
a(n) = binomial(3*n,n)^2 = ( [x^n](1 + x)^(3*n) )^2 = [x^n](F(x)^(9*n)), where F(x) = 1 + x + 4*x^2 + 49*x^3 + 795*x^4 + 15180*x^5 + 320422*x^6 + ... appears to have integer coefficients. For similar results see A000897, A002894, A002897, A006480, A008977 and A186420. - Peter Bala, Jul 12 2016
a(n) ~ 3^(6*n+1)*4^(-2*n-1)/(Pi*n). - Ilya Gutkovskiy, Jul 13 2016
a(n) = Sum_{k=0..n} binomial(n, k)^2*binomial(3*n+k, 2*n). - Seiichi Manyama, Jan 09 2017

A113424 a(n) = (6n)!/((3n)!(2n)!*n!).

Original entry on oeis.org

1, 60, 13860, 4084080, 1338557220, 465817912560, 168470811709200, 62588625639883200, 23717177328413240100, 9124964373613212524400, 3553261127084984957001360, 1397224499394244497967972800, 553883078634868423069470550800, 221068174083308549543680044926400

Offset: 0

Michael Somos, Oct 31 2005

Appears in Ramanujan's theory of elliptic functions of signature 6.
The family of elliptic curves "x=2*H=p^2+q^2-q^3, 0Bradley Klee, Feb 25 2018
The power series with coefficients a(n) * n! plays a central role in the Faber-Zagier relations on the moduli space of algebraic curves; see Pandharipande and Pixton, Section 0.2. - Harry Richman, Aug 19 2024

			G.f. = 1 + 60*x + 13860*x^2 + 4084080*x^3 + 1338557220*x^4 + ... - _Michael Somos_, Dec 02 2018

G. C. Greubel, Table of n, a(n) for n = 0..250
J. Cremona, Elliptic Curves over Q, LMFDB 2017.
Alin Bostan, Armin Straub, and Sergey Yurkevich, On the representability of sequences as constant terms, arXiv:2212.10116 [math.NT], 2022.
H. J. Brothers, Pascal's Prism: Supplementary Material.
S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 23.
Bradley Klee, Geometric G.F. for Ramanujan Periods, seqfans mailing list, 2017.
Bradley Klee, On LMFDB period data, LMFDB-support mailing list, 2018.
Bradley Klee, Weierstrass Solution of Cubic Anharmonic Oscillation, Wolfram Demonstrations Project, 2018.
Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
R. Pandharipande and A. Pixton, Relations in the tautological ring of the moduli space of curves, arXiv:1301.4561 [math.AG], 2020.
S. Ramanujan, Modular Equations and Approximations to Pi, Quarterly Journal of Mathematics, XLV (1914), 350-372.
L. C. Shen, A note on Ramanujan's identities involving the hypergeometric function 2F1(1/6,5/6;1;z), The Ramanujan Journal, 30.2 (2013), 211-222.

a(n) = A347304(6*n)
Elliptic Integrals: A002894, A006480, A000897. Factors: A005809, A066802.
Cf. A188662.

List([0..15],n->Factorial(6*n)/(Factorial(3*n)*Factorial(2*n)*Factorial(n))); # Muniru A Asiru, Apr 08 2018

a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/6, 5/6, 1, 432 x], {x, 0, n}];
Table[Multinomial[n, 2 n, 3 n], {n, 0, 15}] (* Vladimir Reshetnikov, Oct 12 2016 *)
a[ n_] := Multinomial[n, 2 n, 3 n]; (* Michael Somos, Dec 02 2018 *)

{a(n) = if( n<0, 0, (6*n)! / ((3*n)! * (2*n)! * n!))};

G.f.: hypergeometric2F1(1/6, 5/6; 1; 432 * x).
a(n) ~ 432^n/(2*Pi*n). - Ilya Gutkovskiy, Oct 13 2016
a(n) = A005809(n)*A066802(n). - Bradley Klee, Feb 25 2018
0 = a(n)*(-267483013447680*a(n+2) +25577192448000*a(n+3) -204669037440*a(n+4) +372142500*a(n+5)) +a(n+1)*(+408751349760*a(n+2) -57870650880*a(n+3) +546809652*a(n+4) -1088188*a(n+5)) +a(n+2)*(-17884800*a(n+2) +21466920*a(n+3) - 295844*a(n+4) +693*a(n+5)) for all n in Z. - Michael Somos, May 16 2018
From Peter Bala, Feb 28 2020: (Start)
a(n) = C(6*n,2*n)*C(4*n,n).
a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k (apply Mestrovic, equation 39).
(-1)^n*a(n) = [x^(2*n)*y^(2*n)] ( (1 + x + y)*(1 - x + y) )^(4*n).
a(n) = [x^n] ( F(x) )^(60*n), where F(x) = 1 + x + 56*x^2 + 7355*x^3 + 1290319*x^4 + 264117464*x^5 + 59508459679*x^6 + ... appears to have integer coefficients. We conjecture that for k >= 1 the sequence defined by b_k(n) := [x^n] F(x)^(k*n) satisfies the above supercongruences for primes p >= 7. (End)
From Peter Bala, Mar 20 2022: (Start)
Right-hand side of the following identities valid for n >= 1:
Sum_{k = 0..2*n} 4*n*(4*n+k-1)!/(k!*n!*(3*n)!) = (6*n)!/((3*n)!*(2*n)!*n!);
Sum_{k = 0..3*n} 3*n*(3*n+k-1)!/(k!*n!*(2*n)!) = (6*n)!/((3*n)!(2*n)!*n!).
Cf. A001451. (End)
From Peter Bala, Feb 26 2023: (Start)
a(n) = (4^n/n!^2) * Product_{k = n..3*n-1} 2*k + 1.
a(n) = (12^n/n!^2) * Product_{k = 0..n-1} (6*k + 1)*(6*k + 5). (End)
a(n) = 12*(6*n - 1)*(6*n - 5)*a(n-1)/n^2. - Neven Sajko, Jul 19 2023
From Karol A. Penson, Dec 26 2023: (Start)
a(n) = Integral_{x=0..432} x^n*W(x) dx, n>=0, where W(x) = sqrt(18)*MeijerG([[], [0, 0]], [[-1/6, -5/6], []], x/432)/(1296*Pi), where MeijerG is the Meijer G - function.
Apparently, W(x) cannot be represented by any other function. W(x) is positive on x = [0, 432], it diverges at x=0, and monotonically decreases for x>0. It appears that at x=432, W(x) tends to a constant value close to 0.000368414. This integral representation as the n-th power moment of the positive function W(x) on the interval [0, 432] is unique, as W(x) is the solution of the Hausdorff moment problem. (End)
W(x) can be represented in terms of two 2F1 hypergeometric functions, W(x) = hypergeom([1/6, 1/6], [1/3], x/432)/(6*sqrt(Pi)*Gamma(2/3)*Gamma(5/6)*x^(5/6)) - Gamma(2/3)*Gamma(5/6)*sqrt(3)*hypergeom([5/6, 5/6], [5/3], x/432)/(1152*Pi^(5/2)*x^(1/6)), x on (0, 432). - Karol A. Penson, May 16 2025

A008979 a(n) = (6n)!/(n!)^6.

Original entry on oeis.org

1, 720, 7484400, 137225088000, 3246670537110000, 88832646059788350720, 2670177736637149247308800, 85722533226982363751829504000, 2889253496242619386328267523990000, 101097362223624462291180422369532000000, 3644153415887633116359073848179365185734400, 134567406165969006655507763343147223231094784000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
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. A000897, A000984, A002894, A002897, A006480, A008977, A008978, A071549, A071550, A071551, A071552, A186420, A188662.

[Factorial(6*n)/Factorial(n)^6: n in [0..20]]; // Vincenzo Librandi, Aug 13 2014

seq( (6*n)!/(n!)^6, n=0..20); # G. C. Greubel, Feb 17 2020

Table[(6 n)!/(n)!^6, {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2014 *)

vector(21, n, my(m=n-1); (6*m)!/(m!)^6 ) \\ G. C. Greubel, Feb 17 2020

[factorial(6*n)/factorial(n)^6 for n in (0..20)] # G. C. Greubel, Feb 17 2020

From Peter Bala, Jul 12 2016: (Start)
a(n) = binomial(2*n,n)*binomial(3*n,n)*binomial(4*n,n)*
  binomial(5*n,n)*binomial(6*n,n) = ( [x^n](1 + x)^(2*n) ) * ( [x^n](1 + x)^(3*n) ) * ( [x^n](1 + x)^(4*n) ) * ( [x^n](1 + x)^(5*n) ) * ( [x^n](1 + x)^(6*n) ) = [x^n](F(x)^(720*n)), where F(x) = 1 + x + 4478*x^2 + 53085611*x^3 + 926072057094*x^4 + 19977558181209910*x^5 + 493286693783478576177*x^6 + ... appears to have integer coefficients. For similar results see A000897, A002894, A002897, A006480, A008977, A008978, A186420 and A188662. (End)
a(n) ~ 3^(6*n+1/2)*4^(3*n-1)/(Pi*n)^(5/2). - Ilya Gutkovskiy, Jul 12 2016
From Peter Bala, Feb 14 2020: (Start)
a(m*p^k) == a(m*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers m and k - apply Mestrovic, equation 39, p. 12.
a(n) = [(x*y*z*u*v)^n] (1 + x + y + z + u + v)^(6*n). (End)

A241477 Triangle read by rows, number of orbitals classified with respect to the first zero crossing, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 0, 2, 2, 2, 2, 0, 4, 0, 2, 6, 12, 4, 2, 6, 0, 12, 0, 4, 0, 4, 20, 60, 12, 12, 12, 4, 20, 0, 40, 0, 12, 0, 8, 0, 10, 70, 280, 40, 60, 36, 24, 40, 10, 70, 0, 140, 0, 40, 0, 24, 0, 20, 0, 28, 252, 1260, 140, 280, 120, 120, 120, 60, 140, 28, 252, 0, 504, 0

Offset: 1

Peter Luschny, Apr 23 2014

For the combinatorial definitions see A232500. An orbital w over n sectors has its first zero crossing at k if k is the smallest j such that the partial sum(1<=i<=j, w(i))) = 0, where w(i) are the jumps of the orbital represented by -1, 0, 1.

			[1], [ 1]
[2], [ 0,  2]
[3], [ 2,  2,  2]
[4], [ 0,  4,  0,  2]
[5], [ 6, 12,  4,  2,  6]
[6], [ 0, 12,  0,  4,  0, 4]
[7], [20, 60, 12, 12, 12, 4, 20]

Row sums: A056040.

Cf. A232500.

A241477 := proc(n, k)
  if n = 0 then 1
elif k = 0 then 0
elif irem(n, 2) = 0 and irem(k, 2) = 1 then 0
elif k = 1 then (n-1)!/iquo(n-1,2)!^2
else 2*(n-k)!*(k-2)!/iquo(k,2)/(iquo(k-2,2)!*iquo(n-k,2)!)^2
  fi end:
for n from 1 to 9 do seq(A241477(n, k), k=1..n) od;

T[n_, k_] := Which[n == 0, 1, k == 0, 0, Mod[n, 2] == 0 && Mod[k, 2] == 1,  0, k == 1, (n-1)!/Quotient[n-1, 2]!^2, True, 2*(n-k)!*(k-2)!/Quotient[k, 2]/(Quotient[k-2, 2]!*Quotient[n-k, 2]!)^2];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 20 2018, from Maple *)

def A241477_row(n):
    if n == 0: return [1]
    Z = [0]*n; T = [0] if is_odd(n) else []
    for i in (1..n//2): T.append(-1); T.append(1)
    for p in Permutations(T):
        i = 0; s = p[0]
        while s != 0: i += 1; s += p[i];
        Z[i] += 1
    return Z
for n in (1..9): A241477_row(n)

If n is even and k is odd then T(n, k) = 0 else if k = 1 then T(n, 1) = A056040(n-1) else T(n, k) = 2*A057977(k-2)*A056040(n-k).
T(n, n) = A241543(n).
T(n+1, 1) = A126869(n).
T(2*n, 2*n) = |A002420(n)|.
T(2*n+1, 1) = A000984(n).
T(2*n+1, n+1) = A241530(n).
T(2*n+2, 2) = A028329(n).
T(4*n, 2*n) = |A010370(n)|.
T(4*n, 4*n) = |A024491(n)|.
T(4*n+1, 1) = A001448(n).
T(4*n+1, 2*n+1) = A002894(n).

cp's OEIS Frontend

A008978 a(n) = (5*n)!/(n!)^5.

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A060150 a(0) = 1; for n > 0, binomial(2n-1, n-1)^2.

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A091401 Numbers n such that genus of group Gamma_0(n) is zero.

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A000894 a(n) = (2*n)!*(2*n+1)! /((n+1)! *n!^3).

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A000888 a(n) = (2*n)!^2 / ((n+1)!*n!^3).

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A054474 Number of walks on square lattice that start and end at origin after 2n steps, not touching origin at intermediate stages.

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A000894 a(n) = (2n)!(2n+1)! /((n+1)! n!^3).

A000888 a(n) = (2n)!^2 / ((n+1)!n!^3).

A113424 a(n) = (6n)!/((3n)!(2n)!*n!).