cp's OEIS Frontend

A193153 Numbers of spanning trees in the graph join of C_n and C_n.

%I A193153 #35 Feb 16 2025 08:33:15
%S A193153 1296,82944,9150625,1575296100,391476713761,132821015040000,
%T A193153 59042071787233536,33317165538875522500,23276866101199344597601,
%U A193153 19729668557004748392960000,19950922411933407541569256321,23731310247317631978185581240644
%N A193153 Numbers of spanning trees in the graph join of C_n and C_n.
%C A193153 The graph join is the graph obtained by adding all possible edges between different graphs to the graph union.
%H A193153 Alois P. Heinz, <a href="/A193153/b193153.txt">Table of n, a(n) for n = 3..80</a>
%H A193153 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%p A193153 with(LinearAlgebra):
%p A193153 a:= proc(n) local h, i, M;
%p A193153       M:= Matrix(2*n, shape=symmetric);
%p A193153       for h in [seq(seq([i, j+n], j=1..n), i=1..n),
%p A193153                 seq([[i, 1+(i mod n)], [n+i, n+1+(i mod n)]][], i=1..n)]
%p A193153       do M[h[]]:= -1 od;
%p A193153       for i to 2*n do M[i, i]:= -add(M[i, j], j=1..2*n) od;
%p A193153       Determinant(DeleteColumn(DeleteRow(M, 1), 1))
%p A193153     end:
%p A193153 seq(a(n), n=3..20);  # _Alois P. Heinz_, Jul 17 2011
%t A193153 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 A193153 Table[a[n], {n, 3, 20}] (* _Jean-François Alcover_, Feb 12 2023, after _Alois P. Heinz_ *)
%K A193153 nonn
%O A193153 3,1
%A A193153 _Eric W. Weisstein_, Jul 16 2011
%E A193153 Description corrected by _Eric W. Weisstein_, May 10 2017