This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A267982 #41 Nov 16 2024 21:33:48
%S A267982 0,4,32,300,3136,35280,418176,5153148,65436800,851005584,11284224640,
%T A267982 152054927024,2076911622912,28698821320000,400547241561600,
%U A267982 5639401174441500,80010548981049600,1142928467041798800,16425988397113680000,237364657887402183600
%N A267982 a(n) = 4*n*Catalan(n)^2.
%C A267982 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.
%H A267982 Ralf Steiner, <a href="https://www.researchgate.net/publication/340005810_Beispiele_zur_modifizierten_Wallis-Lambert-Reihe">Beispiele zur modifizierten Wallis-Lambert-Reihe</a> (in German).
%F A267982 a(n) = 4*A268085(n).
%F A267982 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
%e A267982 For n=3, a(3)=300.
%t A267982 Table[CatalanNumber[n]^2 (4 n + 0), {n, 0, 20}]
%o A267982 (PARI) a(n) = 4*n*(binomial(2*n, n)/(n+1))^2; \\ _Michel Marcus_, Jan 24 2016
%o A267982 (Magma) [Catalan(n)^2*4*n: n in [0..20]]; // _Vincenzo Librandi_, Jan 24 2016
%o A267982 (Python)
%o A267982 from __future__ import division
%o A267982 A267982_list, b = [0], 4
%o A267982 for n in range(1,10**2):
%o A267982 A267982_list.append(b)
%o A267982 b = b*4*(n+1)*(2*n+1)**2//(n*(n+2)**2) # _Chai Wah Wu_, Jan 28 2016
%Y A267982 Cf. A013709, A267796, A268085.
%K A267982 nonn,easy
%O A267982 0,2
%A A267982 _Ralf Steiner_, Jan 23 2016
%E A267982 More terms from _Vincenzo Librandi_, Jan 24 2016