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A294486 a(n) = binomial(2*n,n) * (2*n+1)^2. 6
1, 18, 150, 980, 5670, 30492, 156156, 772200, 3719430, 17551820, 81477396, 373173528, 1690097500, 7582037400, 33738060600, 149067936720, 654576544710, 2858667619500, 12423860225700, 53760146239800, 231720014946420, 995238809839560, 4260800401533000 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A271755 A197214 A027182 * A228994 A235397 A252971
KEYWORD
nonn,easy
AUTHOR
Daniel Suteu, Oct 31 2017
STATUS
approved

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Last modified May 27 12:27 EDT 2024. Contains 372858 sequences. (Running on oeis4.)