cp's OEIS Frontend

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.

%I A338527 #14 Feb 16 2025 08:34:00
%S A338527 24,13500,34420736,239148450000,3520397039081472,94458953432730437824,
%T A338527 4179422085120000000000000,283894102615246085842939590912,
%U A338527 28059580711858187192007680000000000,3870669526565955444680027453177986243584
%N 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.
%C A338527 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.
%C A338527 These graphs are cographs.
%H A338527 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 A338527 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%F A338527 a(n) = (n + 1)*(2 n + 2)^n*(2 n + 1)^(2 n - 1).
%e A338527 The adjacency matrix of the graph associated with n = 2 is: (compare A204437)
%e A338527  [0, 1, 0, 0, 0, 1, 1, 1]
%e A338527  [1, 0, 0, 0, 0, 1, 1, 1]
%e A338527  [0, 0, 0, 1, 1, 1, 1, 1]
%e A338527  [0, 0, 1, 0, 1, 1, 1, 1]
%e A338527  [0, 0, 1, 1, 0, 1, 1, 1]
%e A338527  [1, 1, 1, 1, 1, 0, 0, 0]
%e A338527  [1, 1, 1, 1, 1, 0, 0, 0]
%e A338527  [1, 1, 1, 1, 1, 0, 0, 0]
%e A338527 a(2) = 13500 because the graph has 13500 spanning trees.
%t A338527 Table[(n + 1)*(2 n + 2)^n*(2 n + 1)^(2 n - 1), {n, 1, 10}]
%Y A338527 Cf.  A338104, A338109.
%K A338527 nonn
%O A338527 1,1
%A A338527 _Rigoberto Florez_, Nov 07 2020
%E A338527 Offset changed by _Georg Fischer_, Nov 03 2023