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Numbers of directed Hamiltonian paths in the complete tripartite graph K_{n,n,n}.
4

%I #24 Nov 02 2024 09:11:07

%S 6,240,37584,15095808,12420864000,18233911296000,43492335022080000,

%T 157551157218115584000,823642573772373884928000,

%U 5970637844437187690496000000,58120324656942369834270720000000,739968068159742816891489484800000000

%N Numbers of directed Hamiltonian paths in the complete tripartite graph K_{n,n,n}.

%H Zlatko Damijanic, <a href="/A234633/b234633.txt">Table of n, a(n) for n = 1..200</a>

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

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

%F a(n) = n!^3 * A110706(n). - _Andrew Howroyd_, May 24 2017

%t Table[2 n!^3 (Binomial[2 n + 1, n + 1] HypergeometricPFQ[{1 - n, 1 - n, 1/2 - n/2, -(n/2)}, {1, -(1/2) - n, -n}, 1] + (n - 1) Binomial[2 n, n + 1] HypergeometricPFQ[{1 - n, 2 - n, 1/2 - n/2, 1 - n/2}, {2, 1/2 - n, -n}, 1]), {n, 10}]) (* _Eric W. Weisstein_, May 26 2017 *)

%Y Cf. A234365, A234616, A110706.

%K nonn

%O 1,1

%A _Eric W. Weisstein_, Dec 28 2013

%E a(7)-a(12) from _Andrew Howroyd_, May 24 2017