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A118266 Coefficient of q^n in (1-q)^5/(1-5q); dimensions of the enveloping algebra of the derived free Lie algebra on 5 letters. 4
1, 0, 10, 40, 205, 1024, 5120, 25600, 128000, 640000, 3200000, 16000000, 80000000, 400000000, 2000000000, 10000000000, 50000000000, 250000000000, 1250000000000, 6250000000000, 31250000000000, 156250000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

For n>=5, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5} such that for fixed, different x_1, x_2, x_3, x_4, x_5 in {1,2,...,n} and fixed y_1, y_2, y_3, y_ 4, y_5 in {1,2,3,4,5} we have f(x_i)<>y_i, (i=1,2,3,4,5). - 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..21.

N. Bergeron, C. Reutenauer, M. Rosas and 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)^5/(1-5q) sum( (-1)^k*C(5,k) 5^(n-k); k=0..min(n,5));

a(n) = 1024*5^(n-5) for n>5. - Jean-François Alcover, Dec 10 2018

MAPLE

f:=n->add((-1)^k*binomial(5, k)*5^(n-k), k=0..min(n, 4)): seq(f(i), i=0..15);

MATHEMATICA

a[n_] := If[n<6, {1, 0, 10, 40, 205, 1024}[[n+1]], 1024*5^(n-5)];

Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 10 2018 *)

CROSSREFS

Cf. A001692, A118264, A118265.

Sequence in context: A002066 A061991 A060580 * A054885 A000449 A027274

Adjacent sequences:  A118263 A118264 A118265 * A118267 A118268 A118269

KEYWORD

nonn

AUTHOR

Mike Zabrocki, Apr 20 2006

STATUS

approved

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Last modified May 30 08:04 EDT 2020. Contains 334712 sequences. (Running on oeis4.)