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A360625 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of a complete graph on n vertices, n >= 1, k >= 0.

%I A360625 #29 Feb 16 2025 08:34:04
%S A360625 1,1,1,2,2,1,1,3,8,12,8,3,1,1,4,20,56,84,90,84,56,20,4,1,1,5,40,176,
%T A360625 440,835,1423,1980,1980,1423,835,440,176,40,5,1,1,6,70,441,1616,4600,
%U A360625 11984,26824,46800,63254,70784,70784,63254,46800,26824,11984,4600,1616,441,70,6,1
%N A360625 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of a complete graph on n vertices, n >= 1, k >= 0.
%H A360625 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 A360625 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 A360625 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>.
%e A360625 Triangle begins:
%e A360625   k= 0 1  2   3   4   5    6    7    8    9  10  11  12 13 14 15
%e A360625 n=1: 1 1
%e A360625 n=2: 1 2  2   1
%e A360625 n=3: 1 3  8  12   8   3    1
%e A360625 n=4: 1 4 20  56  84  90   84   56   20    4   1
%e A360625 n=5: 1 5 40 176 440 835 1423 1980 1980 1423 835 440 176 40  5  1
%e A360625 ...
%o A360625 (SageMath) # uses[betti_numbers, LieAlgebraFromGraph from A360571]
%o A360625 def A360625_row(n):
%o A360625     if n == 1: return [1,1]
%o A360625     return betti_numbers(LieAlgebraFromGraph(graphs.CompleteGraph(n)))
%Y A360625 Cf. A360571, A360572.
%K A360625 nonn,tabf
%O A360625 1,4
%A A360625 _Samuel J. Bevins_, Feb 14 2023