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

Author

Daniel Suteu, Oct 31 2017

Keywords

• nonn
• easy

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.

Crossrefs

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

Programs

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

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 *)

PARI

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

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

D-finite with recurrence n*a(n) +2*(2*n-11)*a(n-1) +16*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Nov 22 2024