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.
1, 0, 3, 8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 472392, 1417176, 4251528, 12754584, 38263752, 114791256, 344373768, 1033121304, 3099363912, 9298091736, 27894275208, 83682825624, 251048476872, 753145430616
Offset: 0
Examples
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.
References
- 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.
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..2097
- Nantel Bergeron, Christophe Reutenauer, Mercedes Rosas, and Mike Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082, 2005. See also Canad. J. Math. 60 (2008), no. 2, 266-296.
- Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See pp. 17, 20.
- Index entries for linear recurrences with constant coefficients, signature (3).
Programs
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Maple
f:=n->coeftayl((1-q)^3/(1-3*q),q=0,n):seq(f(i),i=0..15);
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Mathematica
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 *) Join[{1, 0, 3}, LinearRecurrence[{3}, {8}, 24]] (* Jean-François Alcover, Sep 23 2017 *)
Formula
G.f.: (1-x)^3/(1-3x).
a(n) = 3^{n-1}-3^{n-3} for n>=3.
a(n) = A080923(n-1), n>1.
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
For a(n)>=8, a(n+1)=3*a(n). - Harvey P. Dale, Jun 28 2011
Extensions
Formula corrected Mike Zabrocki, Jul 22 2010
Comments