OFFSET
0,1
COMMENTS
The sequence of repeating coefficients [1,-1,-2,-1,1,2] in the sum in the formula section, is equal to the 6th column in A191898. - Mats Granvik, Mar 19 2012
REFERENCES
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.1, p. 20.
A. Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, Prob. Th. and Related Fields 125, 195-224, 2003.
LINKS
J. M. Borwein and R. Girgensohn, Evaluations of binomial series, Aequat. Math. 70 (2005) 25-36.
A. Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, arXiv:math/0206132 [math.PR], 2002.
Ji-Cai Liu, On two congruences involving Franel numbers, arXiv:2002.03650 [math.NT], 2020.
Courtney Moen, Infinite series with binomial coefficients, Math. Mag. 64 (1) (1991) 53-55.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, El. J. Combin. Numb. Th. 6 (2006) # A27.
Eric Weisstein's World of Mathematics, Bootstrap Percolation.
Eric Weisstein's World of Mathematics, Central Binomial Coefficient.
FORMULA
Sum[1/n^2/Binomial[2n,n], {n,Infinity}].
Pi^2/18 = A013661/3 = Sum[1/(i+0)^2 - 1/(i+1)^2 - 2/(i+2)^2 - 1/(i+3)^2 + 1/(i+4)^2 + 2/(i+5)^2, {i =1, 7, 13, 19, 25,.. infinity, stride of 6}]. - Mats Granvik, Mar 19 2012
Equals Sum_{k>=1} (H(k) - 2*H(2k))/((-3^k)*k). See Liu. - Michel Marcus, Feb 11 2020
Equals Sum_{k>=1} A007814(k)/k^2. - Amiram Eldar, Jul 13 2020
Equals (2/9) * Sum_{k>=0} (-1)^k*(7*k+5)*k!^3/((2*k+1)*(3*k+2)!) [Gosper 1974] - R. J. Mathar, Feb 07 2024
Continued fraction expansion: 1/(2 - 2/(13 - 48/(34 - 270/(65 - ... - 2*(2*n - 1)*n^3/(5*n^2 + 6*n + 2 - ... ))))). See A130549. - Peter Bala, Feb 16 2024
EXAMPLE
0.548311355616075478824138388882008396406316633735...
MATHEMATICA
RealDigits[Pi^2/18, 10, 120][[1]] (* Harvey P. Dale, Aug 14 2011 *)
PROG
(PARI) Pi^2/18 \\ Charles R Greathouse IV, Mar 20 2012
CROSSREFS
KEYWORD
AUTHOR
Eric W. Weisstein, Jul 21 2003
STATUS
approved