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, 865196, 982840, 1112764, 1255889

Offset

1,4

Comments

Sequence T(n,5) in A360571.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

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 (6,-15,20,-15,6,-1).

Formula

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.

G.f.: x^3*(1 + 5*x - 3*x^2 - 3*x^3 + 4*x^4 - 7*x^5 + 5*x^6 - x^7)/(1 - x)^6. - Andrew Howroyd, Nov 14 2025

Mathematica

A364579[n_] := Which[n <= 2, 0, n == 3, 1, n == 4, 11, True, n*(n*(n*(n*(n+30) - 145) - 270) + 2424)/120 - 28]; Array[A364579, 50] (* or *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 1, 11, 48, 140, 329, 668, 1223, 2074}, 50] (* Paolo Xausa, Jan 23 2026 *)

Prog

(Python)
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

Crossrefs

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

Sequence in context: A024530 A117066 A042984 * A008780 A211058 A239460

Adjacent sequences: A364576 A364577 A364578 * A364580 A364581 A364582

Keyword

nonn,easy

Author

Samuel J. Bevins, Aug 14 2023

Status

approved