OFFSET
0,2
REFERENCES
Bruce C. Berndt, Ramanujan's Notebook, Part I, Springer Verlag, 1985. See p. 289, eq. (iii).
Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Wiley, 1987. See p. 386.
LINKS
Muniru A Asiru, Table of n, a(n) for n = 0..200
Jonathan M. Borwein and Robert M. Corless, Emerging Tools for Experimental Mathematics, The American Mathematical Monthly, Vol. 106, No. 10 (1999), pp. 889-909. See p. 905.
Nikita Gogin and Mika Hirvensalo, On the Moments of Squared Binomial Coefficients, (2020).
Bruno Haible and Thomas Papanikolaou, Fast multiprecision evaluation of series of rational numbers, in: J. P. Buhler (ed.), Algorithmic Number Theory, ANTS 1998, Lecture Notes in Computer Science, Vol. 1423, Springer, Berlin, Heidelberg, 1998, pp. 338-350, alternative link.
FORMULA
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]
G.f.: (1 + 8*x)/(1 - 4*x)^(5/2). - Ilya Gutkovskiy, Jan 23 2018
Sum_{n>=0} (-1)^n/a(n) = Pi^2/6 - 3*log(phi)^2 = A145436. - Amiram Eldar, Oct 19 2020
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
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
MAPLE
seq(binomial(2*n, n) * (2*n + 1)^2, n=0..30); # Muniru A Asiru, Jan 23 2018
MATHEMATICA
Array[Binomial[2 #, #] (2 # + 1)^2 &, 23, 0] (* Michael De Vlieger, Nov 01 2017 *)
PROG
(PARI) a(n) = binomial(2*n, n) * (2*n+1)^2
(GAP) sequence := List([0..10], n-> Binomial(2*n, n) * (2*n + 1)^2); # Muniru A Asiru, Jan 23 2018
(Magma) [Binomial(2*n, n)*(2*n+1)^2: n in [0..30]]; // G. C. Greubel, Aug 25 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Daniel Suteu, Oct 31 2017
STATUS
approved