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a(n) = Product_{i=1..n} (2*n-i)*(2*n+i).
1

%I #22 Sep 08 2022 08:45:13

%S 3,180,30240,9979200,5448643200,4446092851200,5068545850368000,

%T 7694052600858624000,15003402571674316800000,

%U 36548288664598635724800000,108767707065845539917004800000,388300714225068577503707136000000

%N a(n) = Product_{i=1..n} (2*n-i)*(2*n+i).

%C The arithmetic mean of these numbers is 2n.

%C The old definition was "a(n) = product of 2n distinct numbers, n just less than 2n and n just more than 2n."

%H Robert Israel, <a href="/A093434/b093434.txt">Table of n, a(n) for n = 1..194</a>

%F a(n) ~ 3^(3*n+1/2) * n^(2*n) / (2*exp(2*n)). - _Vaclav Kotesovec_, Oct 21 2014

%F From _Robert Israel_, Jan 16 2017: (Start)

%F a(n) = (3*n)!/(2*n!).

%F a(n+1) = 3*(3*n+2)*(3*n+1)*a(n). (End)

%e a(3) = (3*4*5)*(7*8*9).

%p seq((3*n)!/(2*n!), n=1..50); # _Robert Israel_, Jan 16 2017

%t Table[(3n)!/(2n*(n - 1)!), {n, 12}] (* _Robert G. Wilson v_, Apr 24 2004 *)

%o (Magma) [Factorial(3*n)/(2*Factorial(n)): n in [1..15]]; // _Vincenzo Librandi_, Jan 17 2017

%o (PARI) for(n=1,30, print1((3*n)!/(2*n!), ", ")) \\ _G. C. Greubel_, Feb 13 2018

%K nonn

%O 1,1

%A _Amarnath Murthy_, Apr 01 2004

%E More terms from _Robert G. Wilson v_, Apr 24 2004

%E Replaced definition with formula provided by _Vaclav Kotesovec_, Oct 21 2014. - _N. J. A. Sloane_, Jan 17 2017