A130549 Numerators of partial sums for a series for 2*Zeta(2)/3 = (Pi^2)/9.

1, 13, 197, 1105, 9211, 130277, 82987349, 331950131, 16929464521, 29241805241, 3538258509761, 6259995854281, 1057939300471201, 1057939300716589, 51133732870640471, 372975463296151087, 107789908892879155343

Offset: 1

Author

Wolfdieter Lang, Jul 13 2007

Keywords

• nonn
• frac
• easy

Comments

Denominators are given in A130550.

The r(n) = 2*Sum_{j = 1..n} 1/(j^2*binomial(2*j,j)) tend, for n -> infinity, to 2*Zeta(2)/3 = (Pi^2)/9, which is approximately 1.096622711.

A related result is zeta(2) = 3*Sum_{j = 1..n} 1/(j^2*binomial(2*j,j)) + n!^4/(2*n)!*Sum_{j >= 1} 1/( Product_{i = 0..n} (j + i)^2 ) valid for n >= 0. See Wilf, equation 5, p. 191. - Peter Bala, Oct 30 2023

Examples

			Rationals r(n): [1, 13/12, 197/180, 1105/1008, 9211/8400, 130277/118800, ...].
r(3) = 1/(1 - 1/(13 - 48/(34))) = 197/180. - _Peter Bala_, Feb 17 2024
		

References

• L. Berggren, T. Borwein and P. Borwein, Pi: A Source Book, Springer, New York, 1997, p. 687.
• A. van der Poorten, A proof that Euler missed..., reprinted in Pi: A Source Book, pp. 439-447, eq. 2', with a hint for the proof in footnote 4.

Links

• C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.
• Wolfdieter Lang, Rationals and limit.
• A. J. van der Poorten, A proof that Euler missed ... Apery's proof of the irrationality of zeta(3), Math. Intelligencer 1 (1978/1979), 195-203.
• Herbert S. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics & Theoretical Computer Science, Vol 3, No 4 (1999).

Crossrefs

Cf. A112093, A112099, A112100, A112102, A112103, A130550.

Programs

Maple

seq(numer(add(2/(k^2*binomial(2*k, k)), k = 1 .. n)), n = 1 .. 17); # Peter Bala, Mar 03 2015

Mathematica

Table[2*Sum[1/(i^2*Binomial[2*i, i]), {i, 1, n}], {n, 1, 20}] // Numerator
Accumulate[Table[1/(n^2 Binomial[2n,n]),{n,20}]]//Numerator (* Harvey P. Dale, Jan 27 2019 *)

PARI

a(n) = numerator(2*sum(i=1, n, 1/(i^2*binomial(2*i, i)))); \\ Michel Marcus, Mar 10 2016

Formula

a(n) = numerator(r(n)), n>=1, with the rationals r(n) defined above.

Numerator of 2*Sum_{i=1..n} 1/(i^2*C(2*i,i)). - Wolfdieter Lang, Oct 07 2008; edited by Michel Marcus, Mar 10 2016

a(n) = A112093(n) for n >= 2. - Georg Fischer, Nov 03 2018

From Peter Bala, Feb 17 2024: (Start)

The sequences {(2*n)! : n >= 1} and {(2*n)!*r(n) : n >= 1} satisfy the same second-order recurrence u(n) = (5*n^2 - 4*n + 1)*u(n-1) - 2*(n - 1)^3*(2*n - 3)*u(n-2) leading to the continued fraction representations r(n) = 1/(1 - 1/(13 - 48/(34 - 270/(65 - ... - 2*(2*n - 3)*(n - 1)^3/(5*n^2 - 4*n + 1 ))))) and Pi^2/9 = 1/(1 - 1/(13 - 48/(34 - 270/(65 - ... - 2*(2*n - 3)*(n - 1)^3/((5*n^2 - 4*n + 1) - ... ))))). (End)