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%I #36 Sep 08 2022 08:44:28
%S 0,1,9,64,425,2751,17640,112848,722601,4638205,29860259,192831288,
%T 1248973544,8112024844,52820112480,344712308064,2254247833257,
%U 14768735480505,96917273443305,636948624057900,4191706659276675,27618897144488595,182181063882796680
%N a(n) = C(3n,n) - C(2n,n).
%C It appears that, with the exception of n = 49, a(n)== 1 (mod n^2) only if n is prime. (Tested to 10,000.) - _Gary Detlefs_, Aug 06 2013
%H T. D. Noe, <a href="/A000846/b000846.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = Sum_{k = 1..n} binomial(n,k)*binomial(2*n,n-k). - _Vladimir Kruchinin_, Nov 12 2014
%F 2*n*(n-1)*(2*n-1)*(11*n^2-33*n+24)*a(n) -(n-1) *(473*n^4 -1892*n^3 +2561*n^2 -1338*n +216) *a(n-1) +6 *(3*n-5) *(3*n-4) *(2*n-3) *(11*n^2-11*n+2) *a(n-2)=0. - _R. J. Mathar_, May 05 2018
%p seq(binomial(3*n,n)-binomial(2*n,n),n=0..10) ; # _R. J. Mathar_, May 05 2018
%t Table[Binomial[3*n, n] - Binomial[2*n, n], {n, 0, 20}] (* _T. D. Noe_, Jun 20 2012 *)
%o (Magma) [Binomial(3*n,n)-Binomial(2*n,n): n in [0..30]]; // _Vincenzo Librandi_, Nov 12 2014
%o (Python)
%o from math import comb
%o def A000846(n): return comb(3*n,n)-comb(n<<1,n) # _Chai Wah Wu_, Sep 07 2022
%K nonn,easy
%O 0,3
%A _Brendan McKay_
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