A088459 -id:A088459 - cp's OEIS frontend

A361044 Triangle read by rows. T(n, k) is the k-th Lie-Betti number of the friendship (or windmill) graph, for n >= 1.

1, 3, 8, 12, 8, 3, 1, 1, 5, 24, 60, 109, 161, 161, 109, 60, 24, 5, 1, 1, 7, 48, 168, 483, 1074, 1805, 2531, 2886, 2531, 1805, 1074, 483, 168, 48, 7, 1

Offset: 1

Peter Luschny, Mar 01 2023

The triangle is inspired by Samuel J. Bevins's A360571.

The friendship graph is constructed by joining n copies of the cycle graph C_3 at a common vertex. F_1 is isomorphic to C_3 (the triangle graph) and has 3 vertices, F_2 is the butterfly graph and has 5 vertices and if n > 2 then F_n has 2*n + 1 vertices.

			The triangle T(n, k) starts:
[1] 1, 3, 8, 12, 8, 3, 1;
[2] 1, 5, 24, 60, 109, 161, 161, 109, 60, 24, 5, 1;
[3] 1, 7, 48, 168, 483, 1074, 1805, 2531, 2886, 2531, 1805, 1074, 483, 168, 48, 7, 1;

Marco Aldi and Samuel Bevins, L_oo-algebras and hypergraphs, arXiv:2212.13608 [math.CO], 2022. See page 9.
Meera G. Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1.
Eric Weisstein's World of Mathematics, Dutch Windmill Graph.
Wikipedia, Friendship Graph.

Cf. A360571 (path graph), A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A360937 (wheel graph).

from sage.algebras.lie_algebras.lie_algebra import LieAlgebra, LieAlgebras
def BettiNumbers(graph):
    D = {}
    for edge in graph.edges():
        e = "x" + str(edge[0])
        f = "x" + str(edge[1])
        D[(e, f)] = {e + f : 1}
    C = (LieAlgebras(QQ).WithBasis().Graded().FiniteDimensional().
         Stratified().Nilpotent())
    L = LieAlgebra(QQ, D, nilpotent=True, category=C)
    H = L.cohomology()
    d = L.dimension() + 1
    return [H[n].dimension() for n in range(d)]
def A361044_row(n):
    return BettiNumbers(graphs.FriendshipGraph(n))
for n in range(1, 4): print(A361044_row(n))

A364579 Fifth Lie-Betti number of a path graph on n vertices.

Original entry on oeis.org

0, 0, 1, 11, 48, 140, 329, 668, 1223, 2074, 3316, 5060, 7434, 10584, 14675, 19892, 26441, 34550, 44470, 56476, 70868, 87972, 108141, 131756, 159227, 190994, 227528, 269332, 316942, 370928, 431895, 500484, 577373, 663278, 758954

Offset: 1

Samuel J. Bevins, Aug 14 2023

nonn

Sequence T(n,5) in A360571.

Marco Aldi and Samuel Bevins, L_oo-algebras and hypergraphs, arXiv:2212.13608 [math.CO], 2022. See page 9.
Meera Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1.
Eric Weisstein's World of Mathematics, Path Graph.

Cf. A360571 (path graph triangle), A088459 (second Lie-Betti number of path graphs), A361230, A362007.

def A364579_up_to(n):
    values = [0, 0, 1, 11]
    for i in range(5, n+1):
        result = (i**5 + 30*i**4 - 145*i**3 - 270*i**2 + 2424*i - 3360)/120
        values.append(int(result))
    return values

a(1) = a(2) = 0, a(3) = 1, a(4) = 11, a(n) = (n^5 + 30*n^4 - 145*n^3 - 270*n^2 + 2424*n - 3360)/120 for n >= 5.

A364946 Sixth Lie-Betti number of a path graph on n vertices.

Original entry on oeis.org

0, 0, 0, 4, 33, 140, 424, 1039, 2213, 4262, 7606, 12786, 20482, 31532, 46952, 67957, 95983, 132710, 180086, 240352, 316068, 410140, 525848, 666875, 837337, 1041814, 1285382, 1573646, 1912774, 2309532, 2771320, 3306209, 3922979, 4631158, 5441062

Offset: 1

Samuel J. Bevins, Aug 14 2023

Sequence T(n,6) in A360571.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Marco Aldi and Samuel Bevins, L_oo-algebras and hypergraphs, arXiv:2212.13608 [math.CO], 2022. See page 9.
Meera Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1.
Eric Weisstein's World of Mathematics, Path Graph.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A360571 (path graph triangle), A088459 (second Lie-Betti number of path graphs), A361230, A362007, A364579.

LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 0, 4, 33, 140, 424, 1039, 2213, 4262, 7606, 12786}, 50] (* Paolo Xausa, May 28 2024 *)

def A364946_up_to(n):
    values = [0, 0, 0, 4,33]
    for i in range(6, n+1):
        result = (i**6 + 45*i**5 - 125*i**4 - 2865*i**3 + 23524*i**2 - 76740*i + 98640)/720
        values.append(int(result))
    return values

a(1) = a(2) = a(3) = 0, a(4) = 4, a(5) = 33, a(n) = (n^6 + 45*n^5 - 125*n^4 - 2865*n^3 + 23524*n^2 - 76740*n + 98640)/720 for n >= 6.

G.f.: x^4*(4 + 5*x - 7*x^2 - 3*x^3 - 4*x^4 + 15*x^5 - 15*x^6 + 7*x^7 - x^8)/(1 - x)^7. - Stefano Spezia, Aug 29 2023

A361014 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the hypercube graph on 2^(n-1) vertices, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 4, 14, 25, 28, 25, 14, 4, 1, 1, 8, 64, 258, 986, 2870, 6134, 11586, 18830, 23832, 25078, 23832, 18830, 11586, 6134, 2870, 986, 258, 64, 8, 1

Offset: 1

Samuel J. Bevins, Feb 28 2023

			Triangle begins:
   k=0 1  2   3   4    5    6     7     8     9    10    11    12    13   14
n=1: 1 1
n=2: 1 2  2   1
n=3: 1 4 14  25  28   25   14     4     1
n=4: 1 8 64 258 986 2870 6134 11586 18830 23832 25078 23832 18830 11586 6134
...

Marco Aldi and Samuel Bevins, L_oo-algebras and hypergraphs, arXiv:2212.13608 [math.CO], 2022. See page 9.
Meera Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1.
Eric Weisstein's World of Mathematics, Hypercube Graph.

Cf. A360571 (path graph), A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A360937 (wheel graph).

# uses[betti_numbers, LieAlgebraFromGraph from A360571]
def A360936_row(n):
    if n == 1: return [1, 1]
    return betti_numbers(LieAlgebraFromGraph(graphs.CubeGraph(n-1)))

A368135 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the Fibonacci trees of order n >= 2.

Original entry on oeis.org

1, 2, 2, 1, 1, 4, 11, 16, 16, 11, 4, 1, 1, 7, 33, 95, 212, 344, 444, 444, 344, 212, 95, 33, 7, 1, 1, 12, 90, 454, 1780, 5489, 14036, 29804, 54007, 83404, 111361, 128378, 128378, 111361, 83404, 54007, 29804, 14036, 5489, 1780, 454, 90, 12, 1

Offset: 2

Samuel J. Bevins, Jan 11 2024

			Triangle begins:
  k=0 1  2  3   4   5    6    7    8    9    10    11   12    13   14   15
n=2: 1 2   2  1
n=3: 1 4  11  16   16   11     4     1
n=4: 1 7  33  95  212  344   444   444   344   212     95     33      7      1
n=5: 1 12 90 454 1780 5489 14036 29804 54007 83404 111361 128378 128378 111361 83404 54007 ...

Marco Aldi and Samuel Bevins, 2-step Nilpotent L_oo-algebras and Hypergraphs, arXiv:2212.13608 [math.CO], 2023. See page 9.
Meera Mainkar, Graphs and two step nilpotent Lie algebras, arXiv:1310.3414 [math.DG], 2013. See page 1.
SageMath Graph Theory, Various families of graphs, see FibonacciTree().

Cf. A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360938 (ladder graph), A360937 (wheel graph).

from sage.algebras.lie_algebras.lie_algebra import LieAlgebra, LieAlgebras
def BettiNumbers(graph):
    D = {}
    for edge in graph.edges():
        e = "x" + str(edge[0])
        f = "x" + str(edge[1])
        D[(e, f)] = {e + f : 1}
    C = (LieAlgebras(QQ).WithBasis().Graded().FiniteDimensional().
         Stratified().Nilpotent())
    L = LieAlgebra(QQ, D, nilpotent=True, category=C)
    H = L.cohomology()
    d = L.dimension() + 1
    return [H[n].dimension() for n in range(d)]
# Example usage:
n = 5
X = BettiNumbers(graphs.FibonacciTree(n))

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A361044 Triangle read by rows. T(n, k) is the k-th Lie-Betti number of the friendship (or windmill) graph, for n >= 1.

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A364579 Fifth Lie-Betti number of a path graph on n vertices.

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A364946 Sixth Lie-Betti number of a path graph on n vertices.

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A361014 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the hypercube graph on 2^(n-1) vertices, n >= 1, k >= 0.

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A368135 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the Fibonacci trees of order n >= 2.

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