A294486 a(n) = binomial(2*n,n) * (2*n+1)^2.

1, 18, 150, 980, 5670, 30492, 156156, 772200, 3719430, 17551820, 81477396, 373173528, 1690097500, 7582037400, 33738060600, 149067936720, 654576544710, 2858667619500, 12423860225700, 53760146239800, 231720014946420, 995238809839560, 4260800401533000

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

a(n) = A000984(n) * A016754(n).

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

Cf. A000984, A002878, A006752, A016754, A145436, A361719.

Sequence in context: A271755 A197214 A027182 * A228994 A235397 A252971

Adjacent sequences: A294483 A294484 A294485 * A294487 A294488 A294489

Keyword

nonn,easy

Author

Daniel Suteu, Oct 31 2017

Status

approved