cp's OEIS Frontend

A118264 Coefficient of q^n in (1-q)^3/(1-3q); dimensions of the enveloping algebra of the derived free Lie algebra on 3 letters.

%I A118264 #34 Jan 03 2025 12:22:51
%S A118264 1,0,3,8,24,72,216,648,1944,5832,17496,52488,157464,472392,1417176,
%T A118264 4251528,12754584,38263752,114791256,344373768,1033121304,3099363912,
%U A118264 9298091736,27894275208,83682825624,251048476872,753145430616
%N A118264 Coefficient of q^n in (1-q)^3/(1-3q); dimensions of the enveloping algebra of the derived free Lie algebra on 3 letters.
%C A118264 a(n) is the number of generalized compositions of n when there are i^2-1 different types of i, (i=1,2,...). - _Milan Janjic_, Sep 24 2010
%D A118264 C. Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. xviii+269 pp.
%H A118264 Michael De Vlieger, <a href="/A118264/b118264.txt">Table of n, a(n) for n = 0..2097</a>
%H A118264 Nantel Bergeron, Christophe Reutenauer, Mercedes Rosas, and Mike Zabrocki, <a href="https://arxiv.org/abs/math/0502082">Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables</a>, arXiv:math.CO/0502082, 2005. See also <a href="https://doi.org/10.4153/CJM-2008-013-4 ">Canad. J. Math.</a> 60 (2008), no. 2, 266-296.
%H A118264 Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, <a href="https://arxiv.org/abs/2412.19670">Conjugation, loop and closure invariants of the iterated-integrals signature</a>, arXiv:2412.19670 [math.RA], 2024. See pp. 17, 20.
%H A118264 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (3).
%F A118264 G.f.: (1-x)^3/(1-3x).
%F A118264 a(n) = 3^{n-1}-3^{n-3} for n>=3.
%F A118264 a(n) = A080923(n-1), n>1.
%F A118264 If p[i]=i^2-1 and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. - _Milan Janjic_, May 02 2010
%F A118264 For a(n)>=8, a(n+1)=3*a(n). - _Harvey P. Dale_, Jun 28 2011
%e A118264 The enveloping algebra of the derived free Lie algebra is characterized as the intersection of the kernels of all partial derivative operators in the space of non-commutative polynomials, a(0) = 1 since all constants are killed by derivatives, a(1) = 0 since no polys of degree 1 are killed, a(2) = 3 since all Lie brackets [x1,x2], [x1,x3], [x2, x3] are killed by all derivative operators.
%p A118264 f:=n->coeftayl((1-q)^3/(1-3*q),q=0,n):seq(f(i),i=0..15);
%t A118264 CoefficientList[Series[(1-q)^3/(1-3q),{q,0,30}],q] (* or *) Join[{1,0,3}, NestList[3#&,8,30]] (* _Harvey P. Dale_, Jun 28 2011 *)
%t A118264 Join[{1, 0, 3}, LinearRecurrence[{3}, {8}, 24]] (* _Jean-François Alcover_, Sep 23 2017 *)
%Y A118264 Cf. A080923, A027376, A118265, A118266.
%K A118264 nonn,easy
%O A118264 0,3
%A A118264 _Mike Zabrocki_, Apr 20 2006
%E A118264 Formula corrected _Mike Zabrocki_, Jul 22 2010