%I #40 Jul 29 2023 11:58:25

%S 1,15,280,5775,126126,2858856,66512160,1577585295,37978905250,

%T 925166131890,22754499243840,564121960420200,14079683012144400,

%U 353428777651788000,8915829964229105280,225890910734335847055,5744976449471863238250,146603287914300510042750

%N a(n) = (1/6)*multinomial(3*n;n,n,n).

%C Number of ways of dividing 3n labeled items into 3 unlabeled boxes with n items in each box.

%C From Antonio Campello (campello(AT)ime.unicamp.br), Nov 11 2009: (Start)

%C A060542(t) is the number of optimal [n,2,d] binary codes that correct at most t errors, i.e., having Hamming distance 2*t + 1 (achieved on length n = 3*t + 2). These codes are all isometric.

%C It is also the number of optimal [n,2,d] binary codes that detect 2*t + 1 errors, i.e., having Hamming distance 2t+2 (obtained by adding an overall parity check to the n = 3*t + 2 optimal codes). These codes are also all isometric.

%C For t = 0, we have the famous MDS, cyclic, simplex code {(000), (101), (110), (011)}. (End)

%C Also the number of distinct adjacency matrices of the complete tripartite graph K_{n,n,n}. - _Eric W. Weisstein_, Apr 21 2017

%H Harry J. Smith, <a href="/A060542/b060542.txt">Table of n, a(n) for n = 1..100</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AdjacencyMatrix.html">Adjacency Matrix</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>

%F a(n) = (3*n)!/((n!)^3*6) = a(n-1)*3*(3*n - 1)*(3*n - 2)/n^2 = A060540(3,n) = A006480(n)/6. - corrected by _Vaclav Kotesovec_, Sep 23 2013

%F a(n) ~ 3^(3*n-1/2)/(4*Pi*n). - _Vaclav Kotesovec_, Sep 23 2013

%p a:= n-> combinat[multinomial](3*n,n$3)/3!:

%p seq(a(n), n=1..18); # _Alois P. Heinz_, Jul 29 2023

%t Table[(3*n)!/(n!^3*6),{n,1,20}] (* _Vaclav Kotesovec_, Sep 23 2013 *)

%t Table[Multinomial[n, n, n], {n, 20}]/6 (* _Eric W. Weisstein_, Apr 21 2017 *)

%o (PARI) { a=1/6; for (n=1, 100, write("b060542.txt", n, " ", a=a*3*(3*n - 1)*(3*n - 2)/n^2); ) } \\ _Harry J. Smith_, Jul 06 2009

%Y Row 3 of A060540.

%Y Cf. A025035.

%K nonn

%O 1,2

%A _Henry Bottomley_, Apr 02 2001

%E Definition revised by _N. J. A. Sloane_, Feb 02 2009