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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 such that its set of indices is a sequence of consecutive integers. 0
1, 0, 1, 4, 22, 144, 1089, 9308, 88562, 927584, 10603178, 131368648, 1753970380, 25112732512, 383925637137, 6243618722124, 107644162715098, 1961478594977856, 37671587406585006, 760654555198989240, 16110333600696417780, 357148428086308848480, 8271374327887650503130 (list; graph; refs; listen; history; text; internal format)



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


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.


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)).



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 */


Cf. A075834.

Sequence in context: A243626 A104991 A027391 * A081002 A222012 A057834

Adjacent sequences:  A134985 A134986 A134987 * A134989 A134990 A134991




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



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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)