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

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, 19098, 20312

Offset

1,3

Comments

Sequence T(n,3) 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 (4,-6,4,-1).

Formula

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

Maple

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

Mathematica

A361230[n_] := If[n <= 2, n-1, n*(n*(n+9) - 40)/6 + 8]; Array[A361230, 50] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 6, 16, 33, 58}, 50] (* Paolo Xausa, Jan 23 2026 *)

Prog

(Python)
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

Crossrefs

Cf. A360571 (path graph triangle).

Sequence in context: A099399 A338164 A118014 * A236773 A131820 A266677

Adjacent sequences: A361227 A361228 A361229 * A361231 A361232 A361233

Keyword

nonn,easy

Author

Samuel J. Bevins, Mar 05 2023

Status

approved