cp's OEIS Frontend

A338110 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 vertices.

%I A338110 #22 Feb 16 2025 08:34:00
%S A338110 1,128,139968,536870912,5000000000000,92442129447518208,
%T A338110 2988151979474457198592,154742504910672534362390528,
%U A338110 12044329605471552321957641846784,1342177280000000000000000000000000000,206097683218942123873399068932507659403264,42281678783395138381516145098915043145456549888
%N A338110 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 vertices.
%C A338110 Equivalently, the graph can be described as the graph on 3*n vertices with labels 0..3*n-1 and with i and j adjacent iff A011655(i + j) = 1.
%C A338110 These graphs are cographs.
%H A338110 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 A338110 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%F A338110 a(n) = n*(2*n)^(3*(n - 1)).
%F A338110 a(n) = A193131(n)/3.
%e A338110 The adjacency matrix of the graph associated with n = 2 is: (compare A204437)
%e A338110   [0, 1, 1, 0, 1, 1]
%e A338110   [1, 0, 0, 1, 1, 0]
%e A338110   [1, 0, 0, 1, 0, 1]
%e A338110   [0, 1, 1, 0, 1, 1]
%e A338110   [1, 1, 0, 1, 0, 0]
%e A338110   [1, 0, 1, 1, 0, 0]
%e A338110 a(2) = 128 because the graph has 128 spanning trees.
%t A338110 Table[n (2 n)^(3 (n - 1)), {n, 1, 10}]
%Y A338110 Cf. A011655, A193131, A204437, A338104.
%K A338110 nonn
%O A338110 1,2
%A A338110 _Rigoberto Florez_, Oct 10 2020