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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.
7
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, 440, 835, 1423, 1980, 1980, 1423, 835, 440, 176, 40, 5, 1, 1, 6, 70, 441, 1616, 4600, 11984, 26824, 46800, 63254, 70784, 70784, 63254, 46800, 26824, 11984, 4600, 1616, 441, 70, 6, 1
OFFSET
1,4
LINKS
M. Aldi and S. Bevins, L_oo-algebras and hypergraphs, arXiv:2212.13608 [math.CO], 2022. See page 9.
M. Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1.
Eric Weisstein's World of Mathematics, Complete Graph.
EXAMPLE
Triangle begins:
k= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
n=1: 1 1
n=2: 1 2 2 1
n=3: 1 3 8 12 8 3 1
n=4: 1 4 20 56 84 90 84 56 20 4 1
n=5: 1 5 40 176 440 835 1423 1980 1980 1423 835 440 176 40 5 1
...
PROG
(SageMath) # uses[betti_numbers, LieAlgebraFromGraph from A360571]
def A360625_row(n):
if n == 1: return [1, 1]
return betti_numbers(LieAlgebraFromGraph(graphs.CompleteGraph(n)))
CROSSREFS
Sequence in context: A331485 A342061 A045995 * A157654 A357437 A078692
KEYWORD
nonn,tabf
AUTHOR
Samuel J. Bevins, Feb 14 2023
STATUS
approved