A362007 -id:A362007 - cp's OEIS frontend

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

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

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

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