This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380149 #43 Jan 21 2025 08:49:27
%S A380149 0,-1215,0,-3189375,0,27348890625,978447237120,15920336210625,
%T A380149 163074539520000,1214314872035265,7134511104000000,34856907746165505,
%U A380149 146828238520320000,547377978676010625,1841813423998894080,5678883183381890625,16238028554439229440,43474602051830210625,109846357522513920000
%N A380149 Characteristic polynomial of the tesseract graph: a(n) = n^6*(n^2-16)*(n^2-4)^4.
%C A380149 Given that the eigenvalues of the adjacency matrix of the tesseract graph are: {4,2,0,-2,-4} and their multiplicities are defined by binomial(4,k) for k = 0..4 which results in {1,4,6,4,1}, and the characteristic polynomial is given by P(x) = Prod_{k=1..j} (x-lambda_k)^m_k with j=5, substitute the eigenvalues and their multiplicities as:
%C A380149 k | eigenvalue(lambda_k) | multiplicity(m_k) | term
%C A380149 ---+----------------------+-------------------+---------
%C A380149 1 | 4 | 1 | (x-4)^1
%C A380149 2 | 2 | 4 | (x-2)^4
%C A380149 3 | 0 | 6 | (x)^6
%C A380149 4 | -2 | 4 | (x+2)^4
%C A380149 5 | -4 | 1 | (x+4)^1
%C A380149 This results in the characteristic polynomial:
%C A380149 P(x) = (x-4) * (x-2)^4 * (x)^6 * (x+2)^4 * (x+4).
%C A380149 Also all terms are a(n) = 0 mod 5.
%H A380149 Paolo Xausa, <a href="/A380149/b380149.txt">Table of n, a(n) for n = 0..10000</a>
%H A380149 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CharacteristicPolynomial.html">Characteristic Polynomial</a>.
%H A380149 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TesseractGraph.html">Tesseract Graph</a>.
%H A380149 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphSpectrum.html">Graph Spectrum</a>.
%H A380149 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypercube_graph">Hypercube Graph</a>.
%F A380149 a(n) = -4096*n^6 + 4352*n^8 - 1792*n^10 + 352*n^12 - 32*n^14 + n^16.
%t A380149 A380149[n_] := n^6*(n^2 - 16)*(n^2 - 4)^4; Array[A380149, 20, 0] (* _Paolo Xausa_, Jan 21 2025 *)
%o A380149 (Python)
%o A380149 a = lambda n: (n**6)*(n**2-16)*(n**2-4)**4
%o A380149 print([a(n) for n in range(0,19)])
%Y A380149 Cf. A001014, A028347, A028566.
%K A380149 sign,easy
%O A380149 0,2
%A A380149 _Darío Clavijo_, Jan 13 2025