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
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).
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
KEYWORD
nonn,easy
AUTHOR
Mike Zabrocki, Apr 20 2006
EXTENSIONS
Formula corrected Mike Zabrocki, Jul 22 2010
STATUS
approved