

A118265


Coefficient of q^n in (1q)^4/(14q); 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
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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, 266296.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets


FORMULA

G.f.: (1q)^4/(14q).
a(n) = sum( (1)^k*C(4,k) 4^(nk); k=0..min(n,4)).
a(n) = 81*4^(n4) for n>3.  JeanFrançois 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 noncommutative 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^(nk), 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^(n4)];
Table[a[n], {n, 0, 24}] (* JeanFrançois 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



