A245391 a(n) = 2^n*binomial(2*(n+1), n).

1, 8, 60, 448, 3360, 25344, 192192, 1464320, 11202048, 85995520, 662165504, 5112102912, 39557939200, 306726174720, 2382605107200, 18537602088960, 144438816276480, 1126891074355200, 8802271391907840, 68829791335219200, 538749548542033920, 4220762508660572160

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).

a(n) = A000079(n)*A001791(n+1). - Robert G. Wilson v, Aug 08 2018

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

Cf. A000079, A001791.

Sequence in context: A285391 A001267 A099156 * A254658 A228514 A233666

Adjacent sequences: A245388 A245389 A245390 * A245392 A245393 A245394

Keyword

nonn

Author

Peter Luschny, Nov 30 2014

Status

approved