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%I #42 Jan 27 2024 10:29:46
%S 1,18,150,980,5670,30492,156156,772200,3719430,17551820,81477396,
%T 373173528,1690097500,7582037400,33738060600,149067936720,
%U 654576544710,2858667619500,12423860225700,53760146239800,231720014946420,995238809839560,4260800401533000
%N a(n) = binomial(2*n,n) * (2*n+1)^2.
%D Bruce C. Berndt, Ramanujan's Notebook, Part I, Springer Verlag, 1985. See p. 289, eq. (iii).
%D Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Wiley, 1987. See p. 386.
%H Muniru A Asiru, <a href="/A294486/b294486.txt">Table of n, a(n) for n = 0..200</a>
%H Jonathan M. Borwein and Robert M. Corless, <a href="http://www.jstor.org/stable/2589743">Emerging Tools for Experimental Mathematics</a>, The American Mathematical Monthly, Vol. 106, No. 10 (1999), pp. 889-909. See p. 905.
%H Nikita Gogin and Mika Hirvensalo, <a href="https://pca-pdmi.ru/2020/files/10/GoHi2020ExtAbstract.pdf">On the Moments of Squared Binomial Coefficients</a>, (2020).
%H Bruno Haible and Thomas Papanikolaou, <a href="https://doi.org/10.1007/BFb0054873">Fast multiprecision evaluation of series of rational numbers</a>, in: J. P. Buhler (ed.), Algorithmic Number Theory, ANTS 1998, Lecture Notes in Computer Science, Vol. 1423, Springer, Berlin, Heidelberg, 1998, pp. 338-350, <a href="https://ginac.de/CLN/binsplit.pdf">alternative link</a>.
%F a(n) = A000984(n) * A016754(n).
%F Sum_{n>=0} 1/a(n) = (8*C - Pi*log(2 + sqrt(3)))/3, where C is Catalan's constant, A006752. [Found by Ramanujan. See Berndt, 1985. - _Amiram Eldar_, Jan 27 2024]
%F G.f.: (1 + 8*x)/(1 - 4*x)^(5/2). - _Ilya Gutkovskiy_, Jan 23 2018
%F Sum_{n>=0} (-1)^n/a(n) = Pi^2/6 - 3*log(phi)^2 = A145436. - _Amiram Eldar_, Oct 19 2020
%F a(n) = Sum_{k = 0..2*n+1} (-1)^(n+k+1) * k^2 * binomial(2*n+1,k)^2. Cf. A361719. - _Peter Bala_, Mar 24 2023
%F Sum_{n>=0} A002878(n)/a(n) = (8*G - Pi*log((10+sqrt(50-22*sqrt(5)))/(10-sqrt(50-22*sqrt(5)))))/5, where G is Catalan's constant (A006752) (found by David Bradley, see Borwein and Corless, 1999). - _Amiram Eldar_, Jan 27 2024
%p seq(binomial(2*n, n) * (2*n + 1)^2, n=0..30); # _Muniru A Asiru_, Jan 23 2018
%t Array[Binomial[2 #, #] (2 # + 1)^2 &, 23, 0] (* _Michael De Vlieger_, Nov 01 2017 *)
%o (PARI) a(n) = binomial(2*n, n) * (2*n+1)^2
%o (GAP) sequence := List([0..10], n-> Binomial(2*n, n) * (2*n + 1)^2); # _Muniru A Asiru_, Jan 23 2018
%o (Magma) [Binomial(2*n,n)*(2*n+1)^2: n in [0..30]]; // _G. C. Greubel_, Aug 25 2018
%Y Cf. A000984, A002878, A006752, A016754, A145436, A361719.
%K nonn,easy
%O 0,2
%A _Daniel Suteu_, Oct 31 2017
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