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A355559 The independence polynomial of the n-folded cube graph evaluated at -1.

%I A355559 #20 Oct 13 2022 13:08:02
%S A355559 -1,-3,-1,9,131,253,25607
%N A355559 The independence polynomial of the n-folded cube graph evaluated at -1.
%C A355559 The independence number alpha(G) of a graph is the cardinality of the largest independent vertex set. The n-folded cube has alpha(G) = A058622(n-1). The independence polynomial for the n-folded cube is given by Sum_{k=0..alpha(G)} A355227(n,k)*t^k, meaning that a(n) is the alternating sum of row n of A355227.
%H A355559 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependencePolynomial.html">Independence polynomial</a>
%H A355559 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FoldedCubeGraph.html">Folded cube graph</a>
%e A355559 Row 5 of A355227 is 1, 16, 80, 160, 120, 16. This means the 5-folded cube graph has independence polynomial 1 + 16*t + 80*t^2 + 160*t^3 + 120*t^4 + 16*t^5. Taking the alternating row sum of row 5, or evaluating the polynomial at -1, gives us 1 - 16 + 80 - 160 + 120 - 16 = 9 = a(5).
%o A355559 (Sage) from sage.graphs.independent_sets import IndependentSets
%o A355559 def a(n):
%o A355559     icount=0
%o A355559     for Iset in IndependentSets(graphs.FoldedCubeGraph(n)):
%o A355559         if len(Iset) % 2 == 0:
%o A355559             icount += 1
%o A355559         else:
%o A355559             icount += -1
%o A355559     return icount
%Y A355559 Cf. A058622, A355227, A290888.
%K A355559 sign,more
%O A355559 2,2
%A A355559 _Christopher Flippen_, Jul 06 2022