A360571 -id:A360571 - cp's OEIS frontend

A088459 Triangle read by rows: T(n,k) represents the number of lozenge tilings of an (n,1,n)-hexagon which include the non-vertical tile above the main diagonal starting in position k+1.

1, 1, 1, 2, 2, 1, 1, 3, 6, 6, 3, 1, 1, 4, 12, 18, 18, 12, 4, 1, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1, 1, 7, 42, 126, 315, 525, 700, 700, 525, 315, 126, 42, 7, 1, 1, 8, 56, 196, 588, 1176, 1960, 2450, 2450, 1960, 1176, 588, 196, 56, 8, 1

Offset: 1

Christopher Hanusa (chanusa(AT)washington.edu), Nov 14 2003

Rows are of length 2, 4, 6, 8, 10, 12, ...
T(n,k)= number of symmetric Dyck paths of length 4n and having k peaks. Example: T(2,3)=2 because we have UU*DU*DU*DD and U*DUU*DDU*D, where U=(1,1), D=(1,-1) and * shows the peaks. - Emeric Deutsch, Feb 22 2004
T(n,k) is also the number of nodes at distance k from a specified node in the n-odd graph for k in 1..n-1. - Eric W. Weisstein, Mar 23 2018
T(n,k) seems to be the k-th Lie-Betti number of the star graph on n vertices. See A360571 for additional information and references. - Samuel J. Bevins, Feb 12 2023

			For example, the number of tilings of a 4,1,4 hexagon which includes the non-vertical tile above the main diagonal starting in position 3 is T(4,2)=12.
Triangle T(n, k) begins:
[1] 1,1,
[2] 1,2, 2,  1,
[3] 1,3, 6,  6,   3,   1,
[4] 1,4,12, 18,  18,  12,   4,   1,
[5] 1,5,20, 40,  60,  60,  40,  20,   5,   1,
[6] 1,6,30, 75, 150, 200, 200, 150,  75,  30,   6,  1,
[7] 1,7,42,126, 315, 525, 700, 700, 525, 315, 126, 42,    7,   1,
[8] 1,8,56,196, 588,1176,1960,2450,2450,1960,1176,588,  196,  56,  8, 1,
[9] 1,9,72,288,1008,2352,4704,7056,8820,8820,7056,4704,2352,1008,288,72,9,1

Columns 0-5 are sequences A000012, A000027, A002378, A002411, A006011 and A004302.

Cf. A000984 (row sums).

A088459 := proc(n,k)
    binomial(n,ceil(k/2))*binomial(n-1,floor(k/2)) ;
end proc:
seq(seq(A088459(n,k),k=0..2*n-1),n=1..10) ; # R. J. Mathar, Apr 02 2017

Table[Binomial[n, Ceiling[k/2]] Binomial[n - 1, Floor[k/2]], {n, 10}, {k, 0, 2 n - 1}] // Flatten (* Eric W. Weisstein, Mar 23 2018 *)

T(n, k) = binomial(n, ceiling(k/2))* binomial(n-1, floor(k/2)), n>0 and k=0 to 2n-1.

Edited and extended by Ray Chandler, Nov 17 2003

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.

Original entry on oeis.org

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

Samuel J. Bevins, Feb 12 2023

			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
  ...

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

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

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))

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)))

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

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

A088459 Triangle read by rows: T(n,k) represents the number of lozenge tilings of an (n,1,n)-hexagon which include the non-vertical tile above the main diagonal starting in position k+1.

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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.

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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.

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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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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.

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

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

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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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