cp's OEIS Frontend

A338104 Number of spanning trees in the join of the disjoint union of two complete graphs each on n vertices with the empty graph on n+1 vertices.

%I A338104 #41 Feb 16 2025 08:34:00
%S A338104 1,4,1200,2074464,10883911680,128615328600000,2881502756476710912,
%T A338104 109416128865750000000000,6508595325997684722663161856,
%U A338104 572150341080161420030586961966080,71062412455566037275496151040000000000
%N A338104 Number of spanning trees in the join of the disjoint union of two complete graphs each on n vertices with the empty graph on n+1 vertices.
%C A338104 Equivalently, the graph can be described as the graph on 3*n + 1 vertices with labels 0..3*n and with i and j adjacent iff A011655(i + j) = 1.
%C A338104 These graphs are cographs.
%H A338104 H-Y. Ching, R. Florez, and A. Mukherjee, <a href="https://arxiv.org/abs/2009.02770">Families of Integral Cographs within a Triangular Arrays</a>, arXiv:2009.02770 [math.CO], 2020.
%H A338104 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%F A338104 a(n) = (n + 1)*(2*n)^n*(2*n + 1)^(2*(n - 1)).
%e A338104 The adjacency matrix of the graph associated with n = 2 is: (compare A204437)
%e A338104   [0, 1, 1, 0, 1, 1, 0]
%e A338104   [1, 0, 0, 1, 1, 0, 1]
%e A338104   [1, 0, 0, 1, 0, 1, 1]
%e A338104   [0, 1, 1, 0, 1, 1, 0]
%e A338104   [1, 1, 0, 1, 0, 0, 1]
%e A338104   [1, 0, 1, 1, 0, 0, 1]
%e A338104   [0, 1, 1, 0, 1, 1, 0]
%e A338104 a(2) = 1200 because the graph has 1200 spanning trees.
%t A338104 Table[(n + 1)*(2 n)^n*(2 n + 1)^(2 (n - 1)), {n, 1, 10}]
%Y A338104 Cf. A011655, A204437, A338109.
%K A338104 nonn
%O A338104 0,2
%A A338104 _Rigoberto Florez_, Oct 10 2020