A268085 -id:A268085 - cp's OEIS frontend

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

Offset: 0

Ralf Steiner, Jan 23 2016

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.

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

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

Cf. A013709, A267796, A268085.

[Catalan(n)^2*4*n: n in [0..20]]; // Vincenzo Librandi, Jan 24 2016

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

a(n) = 4*n*(binomial(2*n, n)/(n+1))^2; \\ Michel Marcus, Jan 24 2016

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

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

More terms from Vincenzo Librandi, Jan 24 2016

cp's OEIS Frontend

A267982 a(n) = 4nCatalan(n)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

Extensions