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

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

Offset: 0

Author

Emanuele Munarini, Apr 07 2011

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

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

Programs

Magma

[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);

PARI

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

Python

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

Formula

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