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

2,4

Comments

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

Links

Table of n, a(n) for n=2..24.

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.

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)).

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 *)

Prog

(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 */

Crossrefs

Cf. A075834.

Sequence in context: A104991 A368562 A027391 * A081002 A222012 A057834

Adjacent sequences: A134985 A134986 A134987 * A134989 A134990 A134991

Keyword

nonn

Author

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

Status

approved