A206300 Expand the real root of y^3 - y + x in powers of x, then multiply coefficient of x^n by -4^n to get integers.

-1, 2, 6, 32, 210, 1536, 12012, 98304, 831402, 7208960, 63740820, 572522496, 5209363380, 47915728896, 444799488600, 4161823309824, 39209074920090, 371626340253696, 3541117629057540

Offset: 0

Author

Roger L. Bagula, Feb 05 2012

Keywords

• sign
• easy
• changed

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

• George E. Andrews, Number Theory, 1971, Dover Publications, New York, pp. 41-43.

Crossrefs

Cf. A000108, A048990, A224884 (signed version).

Cf. A085614.

Programs

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}]
Table[2^(2n - 1) Gamma[(3n - 1)/2]/(Gamma[(n + 1)/2]n!), {n, 0, 20}] (* Dixon J. Jones, Jun 24 2021 *)
Table[2^(2n - 1) Pochhammer[(n + 1)/2, (n-1)]/n!, {n, 0, 20}] (* Dixon J. Jones, Jun 24 2021 *)

PARI

-x/serreverse((x*sqrt(1-4*x))) \\ Thomas Baruchel, Jul 02 2018

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)

a(n) ~ 2^(n-1/2) * 3^(3*n/2-1) / (sqrt(Pi) * n^(3/2)). - Amiram Eldar, Sep 01 2025

Extensions

Edited by N. J. A. Sloane, Feb 09 2012