A267982 a(n) = 4*n*Catalan(n)^2.

0, 4, 32, 300, 3136, 35280, 418176, 5153148, 65436800, 851005584, 11284224640, 152054927024, 2076911622912, 28698821320000, 400547241561600, 5639401174441500, 80010548981049600, 1142928467041798800, 16425988397113680000, 237364657887402183600

Offset

0,2

Comments

The series whose terms are the quotients a(n)/A013709(n) (modified (4n+0) Wallis-Lambert-series-1) is convergent to 4*(1-3/Pi). Proof: Both the Wallis-Lambert-series-1=4/Pi-1 and the elliptic Euler-series=1-2/Pi are absolutely convergent series. Thus any linear combination of the terms of these series will be also absolutely convergent to the value of the linear combination of these series - in this case to 4(1-3/Pi). Q.E.D.

Links

Table of n, a(n) for n=0..19.

Ralf Steiner, Beispiele zur modifizierten Wallis-Lambert-Reihe (in German).

Formula

a(n) = 4*A268085(n).

a(n+1) = a(n)*4*(n+1)*(2*n+1)^2/(n*(n+2)^2) for n > 0. - Chai Wah Wu, Jan 28 2016

Example

For n=3, a(3)=300.

Mathematica

Table[CatalanNumber[n]^2 (4 n + 0), {n, 0, 20}]

Prog

(PARI) a(n) = 4*n*(binomial(2*n, n)/(n+1))^2; \\ Michel Marcus, Jan 24 2016
(Magma) [Catalan(n)^2*4*n: n in [0..20]]; // Vincenzo Librandi, Jan 24 2016
(Python)
from __future__ import division
A267982_list, b = [0], 4
for n in range(1, 10**2):
    A267982_list.append(b)
    b = b*4*(n+1)*(2*n+1)**2//(n*(n+2)**2) # Chai Wah Wu, Jan 28 2016

Crossrefs

Cf. A013709, A267796, A268085.

Sequence in context: A256183 A000766 A370099 * A231258 A294592 A197715

Adjacent sequences: A267979 A267980 A267981 * A267983 A267984 A267985

Keyword

nonn,easy,changed

Author

Ralf Steiner, Jan 23 2016

Extensions

More terms from Vincenzo Librandi, Jan 24 2016

Status

approved