cp's OEIS Frontend

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

%I A361230 #26 Mar 02 2025 10:03:18
%S A361230 0,1,6,16,33,58,92,136,191,258,338,432,541,666,808,968,1147,1346,1566,
%T A361230 1808,2073,2362,2676,3016,3383,3778,4202,4656,5141,5658,6208,6792,
%U A361230 7411,8066,8758,9488,10257,11066,11916,12808,13743,14722,15746,16816,17933
%N A361230 Third Lie-Betti number of a path graph on n vertices.
%C A361230 Sequence T(n,3) in A360571.
%H A361230 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 A361230 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 A361230 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PathGraph.html">Path Graph</a>.
%H A361230 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A361230 a(1) = 0, a(2) = 1, a(n) = (n^3 + 9*n^2 - 40*n + 48)/6 for n >= 3.
%F A361230 a(n) = [x^n] (x^2*(-x^4 + x^3 - 2*x^2 + 2*x + 1))/(x - 1)^4. - _Peter Luschny_, Mar 06 2023
%F A361230 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
%p A361230 gf := (x^2*(-x^4 + x^3 - 2*x^2 + 2*x + 1))/(x - 1)^4:
%p A361230 ser := series(gf, x, 50): seq(coeff(ser, x, n), n = 1..48); # _Peter Luschny_, Mar 06 2023
%o A361230 (Python)
%o A361230 def A361230(n):
%o A361230     values = [0,1]
%o A361230     for i in range(3, n+1):
%o A361230         result = (i^3 + 9*i^2 - 40*i + 48)/6
%o A361230         values.append(result)
%o A361230     return values
%Y A361230 Cf. A360571 (path graph triangle).
%K A361230 nonn,easy
%O A361230 1,3
%A A361230 _Samuel J. Bevins_, Mar 05 2023