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%I #22 Sep 08 2022 08:46:15
%S 0,1,8,75,784,8820,104544,1288287,16359200,212751396,2821056160,
%T 38013731756,519227905728,7174705330000,100136810390400,
%U 1409850293610375,20002637245262400,285732116760449700,4106497099278420000,59341164471850545900,861753537765219528000
%N a(n) = Catalan(n)^2*n.
%C The series whose terms are the quotients a(n)/A013709(n) is convergent to 1-3/Pi.(see formula).
%C 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 1-3/Pi. Q.E.D.
%C Apart from inclusion of a(0) the same as A145600. - _R. J. Mathar_, Feb 07 2016
%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 Sum_{n>=0} a(n)/A013709(n) = 1 - 3/Pi (see A089491).
%e For n=3 the a(3)= 75.
%t Table[CatalanNumber[n]^2 n, {n, 0, 20}]
%o (Magma) [Catalan(n)^2*n: n in [0..20]]; // _Vincenzo Librandi_, Jan 26 2016
%o (PARI) a(n) = n*(binomial(2*n, n)/(n+1))^2; \\ _Altug Alkan_, Jan 26 2016
%Y Cf. A000108, A013709.
%K nonn,easy
%O 0,3
%A _Ralf Steiner_, Jan 26 2016
%E Corrected and extended by _Vincenzo Librandi_, Jan 26 2016