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A360572 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the cycle graph on n vertices, n >= 3, 0 <= k <= 2*n.

%I A360572 #43 Feb 16 2025 08:34:04
%S A360572 1,3,8,12,8,3,1,1,4,14,25,28,25,14,4,1,1,5,20,41,70,90,70,41,20,5,1,1,
%T A360572 6,27,68,146,219,238,219,146,68,27,6,1,1,7,35,105,259,449,644,756,644,
%U A360572 449,259,105,35,7,1,1,8,44,152,422,857,1476,2012,2172,2012,1476,857,422,152,44,8,1
%N A360572 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the cycle graph on n vertices, n >= 3, 0 <= k <= 2*n.
%H A360572 M. Aldi and S. Bevins, <a href="https://arxiv.org/abs/2212.13608">L_oo-algebras and hypergraphs</a>, arXiv:2212.13608 [math.CO], 2022. See page 9.
%H A360572 M. Mainkar, <a href="https://arxiv.org/abs/1310.3414">Graphs and two step nilpotent Lie algebras</a>, arXiv:1310.3414 [math.DG], 2013. See page 1.
%H A360572 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CycleGraph.html">Cycle Graph</a>.
%e A360572 Triangle begins:
%e A360572    k=0 1  2   3   4   5    6    7    8    9   10  11  12  13  14 15 16
%e A360572 n=3  1 3  8  12   8   3    1
%e A360572 n=4  1 4 14  25  28  25   14    4    1
%e A360572 n=5  1 5 20  41  70  90   70   41   20    5    1
%e A360572 n=6  1 6 27  68 146 219  238  219  146   68   27   6   1
%e A360572 n=7  1 7 35 105 259 449  644  756  644  449  259 105  35   7   1
%e A360572 n=8  1 8 44 152 422 857 1476 2012 2172 2012 1476 857 422 152  44  8  1
%e A360572   ...
%o A360572 (SageMath) # uses[betti_numbers, LieAlgebraFromGraph from A360571]
%o A360572 def A360936(n):
%o A360572     return betti_numbers(LieAlgebraFromGraph(graphs.CycleGraph(n)))
%Y A360572 Cf. A360571 (path graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A360937 (wheel graph)
%K A360572 nonn,tabf
%O A360572 3,2
%A A360572 _Samuel J. Bevins_, Feb 12 2023