A338527 Number of spanning trees in the join of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.

24, 13500, 34420736, 239148450000, 3520397039081472, 94458953432730437824, 4179422085120000000000000, 283894102615246085842939590912, 28059580711858187192007680000000000, 3870669526565955444680027453177986243584

Offset: 1

Author

Rigoberto Florez, Nov 07 2020

Keywords

• nonn

Comments

Equivalently, the graph can be described as the graph on 3*n + 2 vertices with labels 0..3*n+1 and with i and j adjacent iff i+j> 0 mod 3.

These graphs are cographs.

Examples

			The adjacency matrix of the graph associated with n = 2 is: (compare A204437)
 [0, 1, 0, 0, 0, 1, 1, 1]
 [1, 0, 0, 0, 0, 1, 1, 1]
 [0, 0, 0, 1, 1, 1, 1, 1]
 [0, 0, 1, 0, 1, 1, 1, 1]
 [0, 0, 1, 1, 0, 1, 1, 1]
 [1, 1, 1, 1, 1, 0, 0, 0]
 [1, 1, 1, 1, 1, 0, 0, 0]
 [1, 1, 1, 1, 1, 0, 0, 0]
a(2) = 13500 because the graph has 13500 spanning trees.
		

Links

• H-Y. Ching, R. Florez, and A. Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.
• Eric Weisstein's World of Mathematics, Spanning Tree

Crossrefs

Cf. A338104, A338109.

Programs

Mathematica

Table[(n + 1)*(2 n + 2)^n*(2 n + 1)^(2 n - 1), {n, 1, 10}]

Formula

a(n) = (n + 1)*(2 n + 2)^n*(2 n + 1)^(2 n - 1).

Extensions

Offset changed by Georg Fischer, Nov 03 2023