OFFSET
0,2
COMMENTS
Also coefficients of the series S(u) for which (-sqrt(3u))*S converges to the larger of the two real roots of x^3 - 3ux + 4u for u >= 4. Specifically, S(u)=Sum_{n>=0} a(n)/(27*u)^(n/2). - Dixon J. Jones, Jun 24 2021
REFERENCES
G. E. Andrews, Number Theory, 1971, Dover Publications New York, pp. 41 - 43.
FORMULA
G.f.: -(12*x)/(2*sin(arcsin(216*x^2-1)/3)+1). - Vladimir Kruchinin, Oct 30 2014
G.f.: -x/Revert((x*sqrt(1-4*x))). - Thomas Baruchel, Jul 02 2018
G.f.: - (1/x) * Revert( x*sqrt(c(4*x)) ), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108 and sqrt(c(4*x)) is the g.f. of A048990. - Peter Bala, Mar 05 2020
From Dixon J. Jones, Jun 24 2021: (Start)
a(n) = 2*A085614(n) for n>=1.
a(n) = 2^(2*n - 1) Gamma((3*n - 1)/2)/(Gamma((n + 1)/2)*n!).
a(n) = (2^(2*n - 1)*((n + 1)/2)_(n-1))/n!, where (x)_k is the Pochhammer symbol. (End)
MATHEMATICA
p[x_] = y /. Solve[y^3 - y + x == 0, y][[1]]
b = Table[-4^n*FullSimplify[ExpandAll[SeriesCoefficient[ Series[p[x], {x, 0, 30}], n]]], {n, 0, 30}]
(* From Dixon J. Jones, Jun 24 2021: (Start) *)
Clear[a]; a=Table[2^(2n - 1) Gamma[(3n - 1)/2]/(Gamma[(n + 1)/2]n!), {n, 0, 20}]
Clear[a]; a=Table[2^(2n - 1) Pochhammer[(n + 1)/2, (n-1)]/n!, {n, 0, 20}] (* End *)
PROG
(PARI) -x/serreverse((x*sqrt(1-4*x))) \\ Thomas Baruchel, Jul 02 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Roger L. Bagula, Feb 05 2012
EXTENSIONS
Edited by N. J. A. Sloane, Feb 09 2012
STATUS
approved