login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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.
0
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
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
KEYWORD
nonn
AUTHOR
Paolo Salvatore and Roberto Tauraso, Feb 05 2008, Feb 22 2008
STATUS
approved