cp's OEIS Frontend

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

%I A364946 #18 Feb 16 2025 08:34:06
%S A364946 0,0,0,4,33,140,424,1039,2213,4262,7606,12786,20482,31532,46952,67957,
%T A364946 95983,132710,180086,240352,316068,410140,525848,666875,837337,
%U A364946 1041814,1285382,1573646,1912774,2309532,2771320,3306209,3922979,4631158,5441062
%N A364946 Sixth Lie-Betti number of a path graph on n vertices.
%C A364946 Sequence T(n,6) in A360571.
%H A364946 Paolo Xausa, <a href="/A364946/b364946.txt">Table of n, a(n) for n = 1..10000</a>
%H A364946 Marco Aldi and Samuel Bevins, <a href="https://arxiv.org/abs/2212.13608">L_oo-algebras and hypergraphs</a>, arXiv:2212.13608 [math.CO], 2022. See page 9.
%H A364946 Meera Mainkar, <a href="https://arxiv.org/abs/1310.3414">Graphs and two step nilpotent Lie algebras</a>, arXiv:1310.3414 [math.DG], 2013. See page 1.
%H A364946 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathGraph.html">Path Graph</a>.
%H A364946 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A364946 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.
%F A364946 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
%t A364946 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 *)
%o A364946 (Python)
%o A364946 def A364946_up_to(n):
%o A364946     values = [0, 0, 0, 4,33]
%o A364946     for i in range(6, n+1):
%o A364946         result = (i**6 + 45*i**5 - 125*i**4 - 2865*i**3 + 23524*i**2 - 76740*i + 98640)/720
%o A364946         values.append(int(result))
%o A364946     return values
%Y A364946  Cf. A360571 (path graph triangle), A088459 (second Lie-Betti number of path graphs), A361230, A362007, A364579.
%K A364946 nonn,easy
%O A364946 1,4
%A A364946 _Samuel J. Bevins_, Aug 14 2023