cp's OEIS Frontend

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.

%I A118266 #22 Jan 03 2025 12:22:27
%S A118266 1,0,10,40,205,1024,5120,25600,128000,640000,3200000,16000000,
%T A118266 80000000,400000000,2000000000,10000000000,50000000000,250000000000,
%U A118266 1250000000000,6250000000000,31250000000000,156250000000000
%N 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.
%C A118266 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
%D A118266 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.
%H A118266 Michael De Vlieger, <a href="/A118266/b118266.txt">Table of n, a(n) for n = 0..1431</a>
%H A118266 Nantel Bergeron, Christophe Reutenauer, Mercedes Rosas, and Mike Zabrocki, <a href="https://arxiv.org/abs/math/0502082">Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables</a>, arXiv:math.CO/0502082, 2005. See also <a href="https://doi.org/10.4153/CJM-2008-013-4 ">Canad. J. Math.</a> 60 (2008), no. 2, 266-296.
%H A118266 Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, <a href="https://arxiv.org/abs/2412.19670">Conjugation, loop and closure invariants of the iterated-integrals signature</a>, arXiv:2412.19670 [math.RA], 2024. See p. 21.
%H A118266 Milan Janjic, <a href="https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>
%F A118266 G.f.: (1-q)^5/(1-5q) sum( (-1)^k*C(5,k) 5^(n-k); k=0..min(n,5));
%F A118266 a(n) = 1024*5^(n-5) for n>5. - _Jean-François Alcover_, Dec 10 2018
%p A118266 f:=n->add((-1)^k*binomial(5,k)*5^(n-k),k=0..min(n,4)): seq(f(i),i=0..15);
%t A118266 a[n_] := If[n<6, {1, 0, 10, 40, 205, 1024}[[n+1]], 1024*5^(n-5)];
%t A118266 Table[a[n], {n, 0, 21}] (* _Jean-François Alcover_, Dec 10 2018 *)
%Y A118266 Cf. A001692, A118264, A118265.
%K A118266 nonn
%O A118266 0,3
%A A118266 _Mike Zabrocki_, Apr 20 2006