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A355226 Irregular triangle read by rows where T(n,k) is the number of independent sets of size k in the n-halved cube graph.

%I A355226 #17 Feb 26 2024 15:39:22
%S A355226 1,1,1,2,1,4,1,8,4,1,16,40,1,32,256,480,120,1,64,1344,11200,36400,
%T A355226 40320,13440,1920,240,1,128,6336,156800,2104480,15644160,63672000,
%U A355226 136970880,147748560,76396800,21087360,4273920,840000,161280,28800,3840,240
%N A355226 Irregular triangle read by rows where T(n,k) is the number of independent sets of size k in the n-halved cube graph.
%C A355226 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)} T(n,k)*t^k.
%C A355226 Since 0 <= k <= alpha(G), row n has length A005864(n) + 1.
%H A355226 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IndependencePolynomial.html">Independence polynomial</a>
%H A355226 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HalvedCubeGraph.html">Halved cube graph</a>
%e A355226 Triangle begins:
%e A355226     k = 0   1   2
%e A355226 n = 1:  1,  1
%e A355226 n = 2:  1,  2
%e A355226 n = 3:  1,  4
%e A355226 n = 4:  1,  8,  4
%e A355226 n = 5:  1, 16, 40
%e A355226 The 4-halved cube graph has independence polynomial 1 + 8*t + 4*t^2.
%o A355226 (Sage) from sage.graphs.independent_sets import IndependentSets
%o A355226 from collections import Counter
%o A355226 def row(n):
%o A355226     if n == 1:
%o A355226         g = graphs.CompleteGraph(1)
%o A355226     else:
%o A355226         g = graphs.HalfCube(n)
%o A355226     setCounts = Counter()
%o A355226     for Iset in IndependentSets(g):
%o A355226         setCounts[len(Iset)] += 1
%o A355226     outList = [0] * len(setCounts)
%o A355226     for n in range(0,len(setCounts)):
%o A355226         outList[n] = setCounts[n]
%o A355226     return outList
%Y A355226 Row sums are A288943.
%Y A355226 Cf. A005864, A355558.
%K A355226 nonn,tabf
%O A355226 1,4
%A A355226 _Christopher Flippen_, Jun 24 2022