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

0,3

Comments

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

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

Table of n, a(n) for n=0..26.

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082 , Canad. J. Math. 60 (2008), no. 2, 266-296.

Index entries for linear recurrences with constant coefficients, signature (3).

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

Example

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.

Maple

f:=n->coeftayl((1-q)^3/(1-3*q), q=0, n):seq(f(i), i=0..15);

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 *)

Crossrefs

Cf. A080923, A027376, A118265, A118266.

Sequence in context: A052855 A133787 A080923 * A006365 A178543 A188175

Adjacent sequences: A118261 A118262 A118263 * A118265 A118266 A118267

Keyword

nonn,easy

Author

Mike Zabrocki, Apr 20 2006

Extensions

Formula corrected Mike Zabrocki, Jul 22 2010

Status

approved