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

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

Author

Samuel J. Bevins, Aug 14 2023

Keywords

• nonn
• easy

Comments

Sequence T(n,6) in A360571.

Links

• 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).

Crossrefs

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

Programs

Mathematica

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 *)

Python

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

Formula

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