Samuel J. Bevins - cp's OEIS frontend

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

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))

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

A363378 Third Lie-Betti number of a cycle graph on n vertices.

Original entry on oeis.org

12, 25, 41, 68, 105, 152, 210, 280, 363, 460, 572, 700, 845, 1008, 1190, 1392, 1615, 1860, 2128, 2420, 2737, 3080, 3450, 3848, 4275, 4732, 5220, 5740, 6293, 6880, 7502, 8160, 8855, 9588, 10360, 11172, 12025, 12920, 13858

Offset: 3

Samuel J. Bevins, Jun 01 2023

nonn

Sequence T(n,3) in A360572.

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.

Cf. A005581, A054000, A028347, A000027, A360572 (cycle graph triangle)

def A363378(n):
    values = [12,25,41]
    for i in range(6, n+1):
        result = (i*(i+11)*(i-2))/6
        values.append(result)
    return values

a(3) = 12, a(4) = 25, a(5) = 41, a(n) = n*(n+11)*(n-2)/6 for n >= 6.
a(n) = A005581(n-4) + A054000(n-1) + A028347(n-2) + A000027(n) for n >= 6.
a(n) = A106058(n+1) - 2 for n >= 6. - Hugo Pfoertner, Jun 02 2023

A362007 Fourth Lie-Betti number of a path graph on n vertices.

Original entry on oeis.org

0, 0, 3, 16, 48, 107, 203, 347, 551, 828, 1192, 1658, 2242, 2961, 3833, 4877, 6113, 7562, 9246, 11188, 13412, 15943, 18807, 22031, 25643, 29672, 34148, 39102, 44566, 50573, 57157, 64353, 72197, 80726, 89978, 99992, 110808

Offset: 1

Samuel J. Bevins, Apr 05 2023

Sequence T(n,4) 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.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

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

Cf. A011379, A006002, A001105, A056106, A000027, A000332, A000217.

def A362007(n):
    values = [0,0,3]
    for i in range(4, n+1):
        result = (i**4 + 18*i**3 - 97*i**2 + 174*i - 168)/24
        values.append(int(result))
    return values

a(1) = a(2) = 0, a(3) = 3, a(n) = (n^4 + 18*n^3 - 97*n^2 + 174*n - 168)/24 for n >= 4.
a(n) = A011379(n-3) + A006002(n-4) + A001105(n-3) + A056106(n-2) + A000027(n-3) + A000332(n-3) + A000217(n-5) + A000027(n-4) for n >= 5.
From Stefano Spezia, Mar 02 2025: (Start)
G.f.: x^2*(3 + x - 2*x^2 - 3*x^3 + 3*x^4 - x^5)/(1 - x)^5.
E.g.f.: (12*(6 + 4*x + x^2) - exp(x)*(72 - 24*x - 36*x^2 - 28*x^3 - x^4))/24. (End)

a(34) and Python program corrected by Robert C. Lyons, Apr 17 2023

A361230 Third Lie-Betti number of a path graph on n vertices.

Original entry on oeis.org

0, 1, 6, 16, 33, 58, 92, 136, 191, 258, 338, 432, 541, 666, 808, 968, 1147, 1346, 1566, 1808, 2073, 2362, 2676, 3016, 3383, 3778, 4202, 4656, 5141, 5658, 6208, 6792, 7411, 8066, 8758, 9488, 10257, 11066, 11916, 12808, 13743, 14722, 15746, 16816, 17933

Offset: 1

Samuel J. Bevins, Mar 05 2023

Sequence T(n,3) 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.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A360571 (path graph triangle).

gf := (x^2*(-x^4 + x^3 - 2*x^2 + 2*x + 1))/(x - 1)^4:
ser := series(gf, x, 50): seq(coeff(ser, x, n), n = 1..48); # Peter Luschny, Mar 06 2023

def A361230(n):
    values = [0,1]
    for i in range(3, n+1):
        result = (i^3 + 9*i^2 - 40*i + 48)/6
        values.append(result)
    return values

a(1) = 0, a(2) = 1, a(n) = (n^3 + 9*n^2 - 40*n + 48)/6 for n >= 3.
a(n) = [x^n] (x^2*(-x^4 + x^3 - 2*x^2 + 2*x + 1))/(x - 1)^4. - Peter Luschny, Mar 06 2023
E.g.f.: exp(x)*(8 - 5*x + 2*x^2 + x^3/6) - 8 - 3*x - x^2/2. - Stefano Spezia, Mar 02 2025

A360937 Triangle read by rows: T(n, k) is the k-th Lie-Betti number of a wheel graph on n vertices, for n >= 3 and k >= 0.

Original entry on oeis.org

1, 3, 8, 12, 8, 3, 1, 1, 4, 20, 56, 84, 90, 84, 56, 20, 4, 1, 1, 5, 32, 108, 212, 371, 547, 547, 371, 212, 108, 32, 5, 1, 1, 6, 45, 171, 442, 1081, 2025, 2616, 2722, 2616, 2025, 1081, 442, 171, 45, 6, 1, 1, 7, 60, 258, 842, 2489, 5440, 8855, 12955, 16785, 16785, 12955, 8855, 5440, 2489, 842, 258, 60, 7, 1

Offset: 3

Samuel J. Bevins, Feb 26 2023

			Triangle T(n, k) 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 20  56  84   90   84   56   20    4    1
n=5: 1 5 32 108 212  371  547  547  371  212  108   32   5   1
n=6: 1 6 45 171 442 1081 2025 2616 2722 2616 2025 1081 442 171 45  6  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, Wheel Graph.

Cf. A360571 (path graph), A360572 (cycle graph), A088459 (star graph), A360625 (complete graph), A360936 (ladder graph), A361044 (friendship graph).

# uses[betti_numbers, LieAlgebraFromGraph from A360571]
def A360937_row(n):
    return betti_numbers(LieAlgebraFromGraph(graphs.WheelGraph(n)))
for n in range(3, 7): print(A360937_row(n))

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)))

A360936 Triangle read by rows: T(n,k) is the k-th Lie-Betti number of the ladder graph on 2*n vertices, n >= 2, k >= 0.

Original entry on oeis.org

1, 2, 2, 1, 1, 4, 14, 25, 28, 25, 14, 4, 1, 1, 6, 32, 89, 204, 357, 437, 437, 357, 204, 89, 32, 6, 1, 1, 8, 54, 207, 680, 1650, 3201, 5310, 6993, 7508, 6993, 5310, 3201, 1650, 680, 207, 54, 8, 1

Offset: 1

Samuel J. Bevins, Feb 26 2023

			Triangle begins:
   k=0 1  2   3   4    5    6    7    8    9   10   11   12   13  14  15 16
n=1: 1 2  2   1
n=2: 1 4 14  25  28   25   14    4    1
n=3: 1 6 32  89 204  357  437  437  357  204   89   32    6    1
n=4: 1 8 54 207 680 1650 3201 5310 6993 7508 6993 5310 3201 1650 680 207 54
...

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, Ladder Graph.

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

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

Original entry on oeis.org

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

Samuel J. Bevins, Feb 14 2023

			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
...

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.

Cf. A360571, A360572.

# uses[betti_numbers, LieAlgebraFromGraph from A360571]
def A360625_row(n):
    if n == 1: return [1,1]
    return betti_numbers(LieAlgebraFromGraph(graphs.CompleteGraph(n)))

cp's OEIS Frontend

User: Samuel J. Bevins

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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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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A363378 Third Lie-Betti number of a cycle graph on n vertices.

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A362007 Fourth Lie-Betti number of a path graph on n vertices.

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A361230 Third Lie-Betti number of a path graph on n vertices.

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Maple

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A360937 Triangle read by rows: T(n, k) is the k-th Lie-Betti number of a wheel graph on n vertices, for n >= 3 and k >= 0.

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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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