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A362007 Fourth Lie-Betti number of a path graph on n vertices.

%I A362007 #54 Mar 02 2025 10:03:15
%S A362007 0,0,3,16,48,107,203,347,551,828,1192,1658,2242,2961,3833,4877,6113,
%T A362007 7562,9246,11188,13412,15943,18807,22031,25643,29672,34148,39102,
%U A362007 44566,50573,57157,64353,72197,80726,89978,99992,110808
%N A362007 Fourth Lie-Betti number of a path graph on n vertices.
%C A362007 Sequence T(n,4) in A360571.
%H A362007 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 A362007 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 A362007 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathGraph.html">Path Graph</a>.
%H A362007 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A362007 a(1) = a(2) = 0, a(3) = 3, a(n) = (n^4 + 18*n^3 - 97*n^2 + 174*n - 168)/24 for n >= 4.
%F A362007 a(n) = A011379(n-3) + A006002(n-4) + A001105(n-3) + A056106(n-2) + A000027(n-3) + A000332(n-3) + A000217(n-5) + A000027(n-4) for n >= 5.
%F A362007 From _Stefano Spezia_, Mar 02 2025: (Start)
%F A362007 G.f.: x^2*(3 + x - 2*x^2 - 3*x^3 + 3*x^4 - x^5)/(1 - x)^5.
%F A362007 E.g.f.: (12*(6 + 4*x + x^2) - exp(x)*(72 - 24*x - 36*x^2 - 28*x^3 - x^4))/24. (End)
%o A362007 (Python)
%o A362007 def A362007(n):
%o A362007     values = [0,0,3]
%o A362007     for i in range(4, n+1):
%o A362007         result = (i**4 + 18*i**3 - 97*i**2 + 174*i - 168)/24
%o A362007         values.append(int(result))
%o A362007     return values
%Y A362007 Cf. A360571 (path graph triangle), A088459 (second Lie-Betti number of path graphs), A361230.
%Y A362007 Cf. A011379, A006002, A001105, A056106, A000027, A000332, A000217.
%K A362007 nonn,easy
%O A362007 1,3
%A A362007 _Samuel J. Bevins_, Apr 05 2023
%E A362007 a(34) and Python program corrected by _Robert C. Lyons_, Apr 17 2023