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.

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, 6, 27, 68, 146, 219, 238, 219, 146, 68, 27, 6, 1, 1, 7, 35, 105, 259, 449, 644, 756, 644, 449, 259, 105, 35, 7, 1, 1, 8, 44, 152, 422, 857, 1476, 2012, 2172, 2012, 1476, 857, 422, 152, 44, 8, 1

Offset: 3

Author

Samuel J. Bevins, Feb 12 2023

Keywords

• nonn
• tabf

Examples

			Triangle begins:
   k=0 1  2   3   4   5    6    7    8    9   10  11  12  13  14 15 16
n=3  1 3  8  12   8   3    1
n=4  1 4 14  25  28  25   14    4    1
n=5  1 5 20  41  70  90   70   41   20    5    1
n=6  1 6 27  68 146 219  238  219  146   68   27   6   1
n=7  1 7 35 105 259 449  644  756  644  449  259 105  35   7   1
n=8  1 8 44 152 422 857 1476 2012 2172 2012 1476 857 422 152  44  8  1
  ...
		

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, Cycle Graph.

Crossrefs

Cf. A360571 (path graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A360937 (wheel graph)

Programs

SageMath

# uses[betti_numbers, LieAlgebraFromGraph from A360571]
def A360936(n):
    return betti_numbers(LieAlgebraFromGraph(graphs.CycleGraph(n)))