|
%I #34 Feb 12 2023 06:08:05
%S 1296,82944,9150625,1575296100,391476713761,132821015040000,
%T 59042071787233536,33317165538875522500,23276866101199344597601,
%U 19729668557004748392960000,19950922411933407541569256321,23731310247317631978185581240644
%N Numbers of spanning trees in the graph join of C_n and C_n.
%C The graph join is the graph obtained by adding all possible edges between different graphs to the graph union.
%H Alois P. Heinz, <a href="/A193153/b193153.txt">Table of n, a(n) for n = 3..80</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%p with(LinearAlgebra):
%p a:= proc(n) local h, i, M;
%p M:= Matrix(2*n, shape=symmetric);
%p for h in [seq(seq([i, j+n], j=1..n), i=1..n),
%p seq([[i, 1+(i mod n)], [n+i, n+1+(i mod n)]][], i=1..n)]
%p do M[h[]]:= -1 od;
%p for i to 2*n do M[i, i]:= -add(M[i, j], j=1..2*n) od;
%p Determinant(DeleteColumn(DeleteRow(M, 1), 1))
%p end:
%p seq(a(n), n=3..20); # _Alois P. Heinz_, Jul 17 2011
%t a[n_] := Module[{h, i, M}, M = Array[0&, {2n, 2n}]; Do[M[[Sequence@@h]] = M[[Sequence@@Reverse[h]]] = -1, {h, Flatten[Table[{i, j+n}, {i, 1, n}, {j, 1, n}], 1] ~Join~ Flatten[Table[{{i, 1+Mod[i, n]}, {n+i, n+1 + Mod[i, n]}}, {i, 1, n}], 1]}]; For[i = 1, i <= 2n, i++, M[[i, i]] = -Sum[M[[i, j]], {j, 1, 2n}]]; Det[Rest /@ Rest[M]]];
%t Table[a[n], {n, 3, 20}] (* _Jean-François Alcover_, Feb 12 2023, after _Alois P. Heinz_ *)
%K nonn
%O 3,1
%A _Eric W. Weisstein_, Jul 16 2011
%E Description corrected by _Eric W. Weisstein_, May 10 2017
|
