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.

1, 0, 10, 40, 205, 1024, 5120, 25600, 128000, 640000, 3200000, 16000000, 80000000, 400000000, 2000000000, 10000000000, 50000000000, 250000000000, 1250000000000, 6250000000000, 31250000000000, 156250000000000

Offset: 0

Author

Mike Zabrocki, Apr 20 2006

Keywords

• nonn

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

• Michael De Vlieger, Table of n, a(n) for n = 0..1431
• 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 p. 21.
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Crossrefs

Cf. A001692, A118264, A118265.

Programs

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 *)

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