A134988 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.

1, 0, 1, 4, 22, 144, 1089, 9308, 88562, 927584, 10603178, 131368648, 1753970380, 25112732512, 383925637137, 6243618722124, 107644162715098, 1961478594977856, 37671587406585006, 760654555198989240, 16110333600696417780, 357148428086308848480, 8271374327887650503130

Offset: 2

Author

Paolo Salvatore and Roberto Tauraso, Feb 05 2008, Feb 22 2008

Keywords

• nonn

Comments

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

Links

• Francis Brown and Jonas Bergström, Inversion of series and the cohomology of the moduli spaces m_(0,n)^δ, arXiv:0910.0120 [math.AG], 2009.
• Jesse Elliott, Asymptotic expansions of the prime counting function, arXiv:1809.06633 [math.NT], 2018.
• P. Salvatore and R. Tauraso, The Operad Lie is Free, arXiv:0802.3010 [math.QA], 2008.

Crossrefs

Cf. A075834.

Programs

Mathematica

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 *)

PARI

N=66;  x='x+O('x^N);
F = sum(n=0,N,x^n*n!);
gf= x - serreverse(x*F);  Vec(Ser(gf))
/* Joerg Arndt, Mar 07 2013 */

Formula

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

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

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