A267982 - OEIS

%I #38 Sep 08 2022 08:46:15

%S 0,4,32,300,3136,35280,418176,5153148,65436800,851005584,11284224640,

%T 152054927024,2076911622912,28698821320000,400547241561600,

%U 5639401174441500,80010548981049600,1142928467041798800,16425988397113680000,237364657887402183600

%N a(n) = 4*n*Catalan(n)^2.

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

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

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

%o (PARI) a(n) = 4*n*(binomial(2*n, n)/(n+1))^2; \\ _Michel Marcus_, Jan 24 2016

%o (Magma) [Catalan(n)^2*4*n: n in [0..20]]; // _Vincenzo Librandi_, Jan 24 2016 *)

%o (Python)

%o from __future__ import division

%o A267982_list, b = [0], 4

%o for n in range(1,10**2):

%o     A267982_list.append(b)

%o     b = b*4*(n+1)*(2*n+1)**2//(n*(n+2)**2) # _Chai Wah Wu_, Jan 28 2016

%Y Cf. A013709, A267796, A268085.

%K nonn

%O 0,2

%A _Ralf Steiner_, Jan 23 2016

%E More terms from _Vincenzo Librandi_, Jan 24 2016