(((n + 1)!)^2·(2·n)!)/(n!^2·(2·n + 2)!)
= ((n!·(n + 1))^2·(2·n)!)/(n!^2·(2·n)!·(2·n + 1)·(2·n + 2))
= (n!^2·(n + 1)^2·(2·n)!)/(n!^2·(2·n)!·(2·n + 1)·(2·n + 2))
= ((n + 1)^2·(2·n)!)/((2·n)!·(2·n + 1)·(2·n + 2))
= ((n + 1)^2)/((2·n + 1)·(2·n + 2))
= ((n + 1)^2)/((2·n + 1)·2·(n + 1))
= (n + 1)/((2·n + 1)·2)
= (n + 1)/(4·n + 2)