A188662 - OEIS

%I #40 Mar 15 2023 16:27:16

%S 1,9,225,7056,245025,9018009,344622096,13521038400,540917591841,

%T 21966328580625,902702926350225,37456461988358400,1566697064677290000,

%U 65973795093338597136,2794203818390077646400,118933541228935777741056,5084343623375056062840609

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

%C 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

%H Seiichi Manyama, <a href="/A188662/b188662.txt">Table of n, a(n) for n = 0..604</a>

%F a(n) = A005809(n)^2.

%F 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

%F a(n) ~ 3^(6*n+1)*4^(-2*n-1)/(Pi*n). - _Ilya Gutkovskiy_, Jul 13 2016

%F a(n) = Sum_{k=0..n} binomial(n, k)^2*binomial(3*n+k, 2*n). - _Seiichi Manyama_, Jan 09 2017

%t Table[Binomial[3 n, n]^2, {n, 0, 16}]

%o (Maxima) makelist(binomial(3*n,n)^2,n,0,16);

%o (Magma) [Binomial(3*n,n)^2: n in [0..100]]; // _Vincenzo Librandi_, Apr 08 2011

%o (PARI) a(n) = binomial(3*n,n)^2; \\ _Michel Marcus_, Nov 01 2016

%o (Python)

%o from math import comb

%o def A188662(n): return comb(3*n,n)**2 # _Chai Wah Wu_, Mar 15 2023

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

%K nonn,easy

%O 0,2

%A _Emanuele Munarini_, Apr 07 2011