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

Offset: 1

Author

Samuel J. Bevins, Mar 05 2023

Keywords

• nonn
• easy

Comments

Sequence T(n,3) in A360571.

Links

• 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).

Crossrefs

Cf. A360571 (path graph triangle).

Programs

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

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

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