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