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%I #15 Jul 17 2022 23:28:18
%S 0,-1,-3,-3,25,-135,-2079,1879969
%N The independence polynomial of the n-halved cube graph evaluated at -1.
%C The independence number alpha(G) of a graph is the cardinality of the largest independent vertex set. The n-halved graph has alpha(G) = A005864(n). The independence polynomial for the n-halved cube is given by Sum_{k=0..alpha(G)} A355226(n,k)*t^k, meaning that a(n) is the alternating sum of row n of A355226.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependencePolynomial.html">Independence polynomial</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HalvedCubeGraph.html">Halved cube graph</a>
%e Row 5 of A355226 is 1, 16, 40. This means the 5-halved cube graph has independence polynomial 1 + 16*t + 40*t^2. Taking the alternating row sum of row 5, or evaluating the polynomial at -1, gives us 1 - 16 + 40 = 25 = a(5).
%o (Sage) from sage.graphs.independent_sets import IndependentSets
%o def a(n):
%o if n == 1:
%o g = graphs.CompleteGraph(1)
%o else:
%o g = graphs.HalfCube(n)
%o icount=0
%o for Iset in IndependentSets(g):
%o if len(Iset) % 2 == 0:
%o icount += 1
%o else:
%o icount += -1
%o return icount
%Y Cf. A005864, A355226, A288943.
%K sign,more
%O 1,3
%A _Christopher Flippen_, Jul 06 2022
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