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A118265 Coefficient of q^n in (1-q)^4/(1-4q); dimensions of the enveloping algebra of the derived free Lie algebra on 4 letters. 3
1, 0, 6, 20, 81, 324, 1296, 5184, 20736, 82944, 331776, 1327104, 5308416, 21233664, 84934656, 339738624, 1358954496, 5435817984, 21743271936, 86973087744, 347892350976, 1391569403904, 5566277615616, 22265110462464, 89060441849856 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

For n>=4, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4} such that for fixed, different x_1, x_2, x_3, x_4 in {1,2,...,n} and fixed y_1, y_2, y_3, y_ 4 in {1,2,3,4} we have f(x_i)<>y_i, (i=1,2,3,4). - Milan Janjic, May 13 2007

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..24.

N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

FORMULA

G.f.: (1-q)^4/(1-4q).

a(n) = sum( (-1)^k*C(4,k) 4^(n-k); k=0..min(n,4)).

a(n) = 81*4^(n-4) for n>3. - Jean-François Alcover, Dec 10 2018

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) = 6 since all Lie brackets [x1,x2], [x1,x3], [x1, x4], [x2,x3], [x2,x4], [x3,x4] are killed by all derivative operators.

MAPLE

f:=n->add((-1)^k*C(4, k)*4^(n-k), k=0..min(n, 4)); seq(f(i), i=0..15);

MATHEMATICA

a[n_] := If[n<4, {1, 0, 6, 20}[[n+1]], 81*4^(n-4)];

Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Dec 10 2018 *)

CROSSREFS

Cf. A027377, A118264, A118266.

Sequence in context: A240043 A058494 A147979 * A204271 A255469 A226638

Adjacent sequences:  A118262 A118263 A118264 * A118266 A118267 A118268

KEYWORD

nonn

AUTHOR

Mike Zabrocki, Apr 20 2006

STATUS

approved

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Last modified April 25 23:48 EDT 2019. Contains 322465 sequences. (Running on oeis4.)