Laguerre's Repeated Fraction

The Continued Fraction

$${(x+1)^n-(x-1)^n\over (x+1)^n+(x-1)^n}= {n\over x+}{n^2-1\over 3x+}{n^2-2^2\over 5x+\ldots}.$$

References

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 13, 1959.