Behold! Alpha and Beta are on their way!

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Let \(\alpha_{n}\) and \(\beta_{n}\) be the roots of

\[x^2+(2n+1)x+n^2\]

The sum of

\[\frac{1}{(\alpha_{3}+1)(\beta_{3}+1)}+\frac{1}{(\alpha_{4}+1)(\beta_{4}+1)}+\ldots+\frac{1}{(\alpha_{20}+1)(\beta_{20}+1)}\]

can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers.

What is the value of \(b-a\)?

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