Representation of 1 as Sum of n Unit Fractions

Theorem
Let $U \left({n}\right)$ denote the number of different ways of representing $1$ as the sum of $n$ unit fractions.

Then for various $n$, $U \left({n}\right)$ is given by the following table:


 * {| border="1"

! align="right" style = "padding: 2px 10px" | $n$ ! align="right" style = "padding: 2px 10px" | $U \left({n}\right)$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $2$
 * align="right" style = "padding: 2px 10px" | $1$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $3$
 * align="right" style = "padding: 2px 10px" | $4$
 * align="right" style = "padding: 2px 10px" | $14$
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $147$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $3462$
 * }
 * align="right" style = "padding: 2px 10px" | $5$
 * align="right" style = "padding: 2px 10px" | $147$
 * align="right" style = "padding: 2px 10px" | $6$
 * align="right" style = "padding: 2px 10px" | $3462$
 * }
 * }

Proof
Trivially:
 * $1 = \dfrac 1 1$

and it follows that: $U \left({1}\right) = 1$

Also trivially:
 * $1 = \dfrac 1 2 + \dfrac 1 2$

and it follows that: $U \left({2}\right) = 1$

From Sum of 3 Unit Fractions that equals 1:
 * $U \left({3}\right) = 3$

From Sum of 4 Unit Fractions that equals 1:
 * $U \left({4}\right) = 14$

From Sum of 5 Unit Fractions that equals 1:
 * $U \left({5}\right) = 147$