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A193131 Numbers of spanning trees of the complete tripartite graphs K_{n,n,n}.

%I A193131 #26 Feb 16 2025 08:33:15
%S A193131 3,384,419904,1610612736,15000000000000,277326388342554624,
%T A193131 8964455938423371595776,464227514732017603087171584,
%U A193131 36132988816414656965872925540352,4026531840000000000000000000000000000,618293049656826371620197206797522978209792
%N A193131 Numbers of spanning trees of the complete tripartite graphs K_{n,n,n}.
%H A193131 Alois P. Heinz, <a href="/A193131/b193131.txt">Table of n, a(n) for n = 1..50</a>
%H A193131 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>
%H A193131 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%F A193131 a(n) = 3*8^(n-1)*n^(3*n-2).
%p A193131 with(LinearAlgebra):
%p A193131 a:= proc(n) local h, i, M;
%p A193131       M:= Matrix(3*n, shape=symmetric);
%p A193131       for h in [seq(seq([[i, j+n],[i, j+2*n],[i+n, j+2*n]][],
%p A193131                 j=1..n), i=1..n)]
%p A193131       do M[h[]]:= -1 od;
%p A193131       for i to 3*n do M[i, i]:= -add(M[i, j], j=1..3*n) od;
%p A193131       Determinant(DeleteColumn(DeleteRow(M, 1), 1))
%p A193131     end:
%p A193131 seq(a(n), n=1..12);  # _Alois P. Heinz_, Jul 18 2011
%t A193131 Table[3 8^(n - 1) n^(3 n - 2), {n, 11}]
%o A193131 (PARI) a(n)=3*n^(3*n-2)<<(3*n-3) \\ _Charles R Greathouse IV_, Jul 29 2011
%K A193131 nonn,easy
%O A193131 1,1
%A A193131 _Eric W. Weisstein_, Jul 16 2011
%E A193131 More terms from _Alois P. Heinz_, Jul 18 2011