OFFSET
0,2
COMMENTS
The g.f. is the derivative of the REVERT transform of x/(1+2*x)^2. - Thomas Baruchel, Jul 02 2018
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = 2^(3*n+2)*(n+1/2)!/(sqrt(Pi)*(n+2)*n!).
a(n) = (4*(2*n+1)*(n+1))/(n*(n+2))*a(n-1) for n >= 1.
O.g.f.: 4/(sqrt(1-8*x)*(1+sqrt(1-8*x))^2).
From Amiram Eldar, Jan 27 2024: (Start)
Sum_{n>=0} 1/a(n) = 2/7 + 44*arccot(sqrt(7))/(7*sqrt(7)).
Sum_{n>=0} (-1)^n/a(n) = 2/9 + 26*log(2)/27. (End)
MAPLE
a := n -> 2^(3*n+2)*(n+1/2)!/(sqrt(Pi)*(n+2)*n!):
seq(a(n), n = 0..21);
MATHEMATICA
CoefficientList[Series[4/(Sqrt[1 - 8*x]*(1 + Sqrt[1 - 8*x])^2), {x, 0, 50}], x] (* G. C. Greubel, Apr 06 2017 *)
a[n_] := 2^n*Binomial[2 n + 2, n]; Array[a, 22, 0] (* Robert G. Wilson v, Aug 08 2018 *)
PROG
(Sage)
@CachedFunction
def A245391(n):
return (4*(2*n+1)*(n+1))/(n*(n+2))*a(n-1) if n > 0 else 1
[A245391(n) for n in range(22)]
(PARI) my(x='x+O('x^50)); Vec(4/(sqrt(1-8*x)*(1+sqrt(1-8*x))^2)) \\ G. C. Greubel, Apr 06 2017
(PARI) my(x='x+O('x^33)); Vec(deriv(serreverse(x/(1+2*x)^2))) \\ Thomas Baruchel, Jul 02 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Nov 30 2014
STATUS
approved