A134988 - OEIS

%I #25 Apr 28 2019 18:12:02

%S 1,0,1,4,22,144,1089,9308,88562,927584,10603178,131368648,1753970380,

%T 25112732512,383925637137,6243618722124,107644162715098,

%U 1961478594977856,37671587406585006,760654555198989240,16110333600696417780,357148428086308848480,8271374327887650503130

%N Number of formal expressions obtained by applying iterated binary brackets to n indexed symbols x_1, ..., x_n such that: 1) each symbol appears exactly once; 2) the smallest index inside a bracket appears on the left hand side and the largest index appears on the right hand side; 3) the outer bracket is the only bracket whose set of indices is a sequence of consecutive integers.

%C a(n) is the number of generators in arity n of the operad Lie, when considered as a free non-symmetric operad.

%H Francis Brown and Jonas Bergström, <a href="http://arxiv.org/abs/0910.0120">Inversion of series and the cohomology of the moduli spaces m_(0,n)^δ</a>, arXiv:0910.0120 [math.AG], 2009.

%H Jesse Elliott, <a href="https://arxiv.org/abs/1809.06633">Asymptotic expansions of the prime counting function</a>, arXiv:1809.06633 [math.NT], 2018.

%H P. Salvatore and R. Tauraso, <a href="https://arxiv.org/abs/0802.3010">The Operad Lie is Free</a>, arXiv:0802.3010 [math.QA], 2008.

%F a(2) = 1, a(n) = Sum_{k=2..n-2} ((k+1)*a(k+1) + a(k))*a(n-k), n > 2;

%F G.f.: x - series_reversion(x*F(x)), where F(x) is the g.f. of the factorials (A000142).

%F a(n) = (1/e)*(1 - 3/n - 5/(2n^2) + O(1/n^3)).

%t terms = 23; F[x_] = Sum[n! x^n, {n, 0, terms+1}]; CoefficientList[(x - InverseSeries[Series[x F[x], {x, 0, terms+1}], x])/x^2, x] (* _Jean-François Alcover_, Feb 17 2019 *)

%o (PARI)

%o N=66;  x='x+O('x^N);

%o F = sum(n=0,N,x^n*n!);

%o gf= x - serreverse(x*F);  Vec(Ser(gf))

%o /* _Joerg Arndt_, Mar 07 2013 */

%Y Cf. A075834.

%K nonn

%O 2,4

%A Paolo Salvatore and _Roberto Tauraso_, Feb 05 2008, Feb 22 2008