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A130549
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Numerators of partial sums for a series for 2*Zeta(2)/3 = (Pi^2)/9.
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6
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1, 13, 197, 1105, 9211, 130277, 82987349, 331950131, 16929464521, 29241805241, 3538258509761, 6259995854281, 1057939300471201, 1057939300716589, 51133732870640471, 372975463296151087, 107789908892879155343
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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a(n) = numerator(r(n)), n>=1, with the rationals r(n) defined above.
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)
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EXAMPLE
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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
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MAPLE
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seq(numer(add(2/(k^2*binomial(2*k, k)), k = 1 .. n)), n = 1 .. 17); # Peter Bala, Mar 03 2015
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n) = numerator(2*sum(i=1, n, 1/(i^2*binomial(2*i, i)))); \\ Michel Marcus, Mar 10 2016
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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