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A267981 a(n) = Catalan(n)^2*(4n + 2).

%I A267981 #30 Sep 08 2022 08:46:15
%S A267981 2,6,40,350,3528,38808,453024,5521230,69526600,898283672,11848435872,
%T A267981 158966514616,2163449607200,29802622140000,414852500188800,
%U A267981 5827381213589550,82510878636707400,1176544010190087000,16882265852589060000,243611096252860135800
%N A267981 a(n) = Catalan(n)^2*(4n + 2).
%C A267981 Numerator of (4n+2)*(Wallis-Lambert-series-1)(n) with denominator A013709(n) convergent to 2*(1-2/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 2*(1-2/Pi). Q.E.D.
%H A267981 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 A267981 G.f.: (Pi-2*EllipticE(16*x))/(2*Pi*x). - _Benedict W. J. Irwin_, Jul 14 2016
%F A267981 a(n) ~ 4^(2*n+1)/(Pi*n^2). - _Ilya Gutkovskiy_, Jul 14 2016
%F A267981 Recurrence: (n+1)^2*a(n) = 4*(2*n - 1)*(2*n + 1)*a(n-1). - _Vaclav Kotesovec_, Jul 16 2016
%F A267981 Sum_{n>=0} a(n)/2^(4*n+2) = 2 - 4/Pi. - _Vaclav Kotesovec_, Jul 16 2016
%e A267981 For n=3 the a(3)=350.
%t A267981 Table[CatalanNumber[n]^2 (4 n + 2), {n, 0, 20}] (* _Vincenzo Librandi_, Jan 25 2016 *)
%o A267981 (Magma) [Catalan(n)^2*(4*n+2):n in [0..20]]; // _Vincenzo Librandi_, Jan 25 2016
%o A267981 (PARI) a000108(n) = binomial(2*n, n)/(n+1)
%o A267981 a(n) = a000108(n)^2 * (4*n+2) \\ _Felix Fröhlich_, Jul 14 2016
%Y A267981 Cf. A013709 (denominator). Equals twice A000891.
%K A267981 nonn,frac
%O A267981 0,1
%A A267981 _Ralf Steiner_, Jan 23 2016
%E A267981 More terms from _Vincenzo Librandi_, Jan 25 2016