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A193153
Numbers of spanning trees in the graph join of C_n and C_n.
3
1296, 82944, 9150625, 1575296100, 391476713761, 132821015040000, 59042071787233536, 33317165538875522500, 23276866101199344597601, 19729668557004748392960000, 19950922411933407541569256321, 23731310247317631978185581240644
OFFSET
3,1
COMMENTS
The graph join is the graph obtained by adding all possible edges between different graphs to the graph union.
LINKS
Eric Weisstein's World of Mathematics, Spanning Tree
MAPLE
with(LinearAlgebra):
a:= proc(n) local h, i, M;
M:= Matrix(2*n, shape=symmetric);
for h in [seq(seq([i, j+n], j=1..n), i=1..n),
seq([[i, 1+(i mod n)], [n+i, n+1+(i mod n)]][], i=1..n)]
do M[h[]]:= -1 od;
for i to 2*n do M[i, i]:= -add(M[i, j], j=1..2*n) od;
Determinant(DeleteColumn(DeleteRow(M, 1), 1))
end:
seq(a(n), n=3..20); # Alois P. Heinz, Jul 17 2011
MATHEMATICA
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]]];
Table[a[n], {n, 3, 20}] (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A265478 A017344 A223689 * A223273 A017464 A017596
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Jul 16 2011
EXTENSIONS
Description corrected by Eric W. Weisstein, May 10 2017
STATUS
approved