Combinatorial Numbers

Let there be a recurrence relation ${ P }_{ a }\left( n \right) =n\cdot { P }_{ a }\left( n-1 \right) +{ a }^{ n }$ , where ${ P }_{ a }\left( 0 \right) =1$. Show that ${ P }_{ a }\left( n \right) \sim { e }^{ a }n!$ for large $n$.

Solution

We first show that the above recurrence relation constructs the sum ${P}_{a}(n) = n!\left( \sum _{ k=0 }^{ n }{ \frac { { a }^{ k } }{ k! } } \right).$

Now we prove by induction.

When $k=1$

$1! \left(\frac{{a}^{0}}{0!} + \frac{{a}^{1}}{1!}\right) = 1(1)+a$

$1!\left( \sum _{ k=0 }^{ 1 }{ \frac { { a }^{ k } }{ k! } } \right) = 1\cdot {P}_{a}(0) + {a}^{1} = {P}_{a}(1).$

When $k=n$

$n! \left(\frac{{a}^{0}}{0!} + \frac{{a}^{1}}{1!} + ...+ \frac{{a}^{n}}{n!}\right) = n(n-1)! \left(\frac{{a}^{0}}{0!} + \frac{{a}^{1}}{1!} + ...+ \frac{{a}^{n-1}}{(n-1)!}\right) + {a}^{n}$

$n!\left( \sum _{ k=0 }^{ n }{ \frac { { a }^{ k } }{ k! } } \right) = n\cdot {P}_{a}(n-1) + {a}^{n-1} = {P}_{a}(n).$

When $k=n+1$

$(n+1)! \left(\frac{{a}^{0}}{0!} + \frac{{a}^{1}}{1!} + ...+ \frac{{a}^{n+1}}{(n+1)!}\right) = (n+1)(n)! \left(\frac{{a}^{0}}{0!} + \frac{{a}^{1}}{1!} + ...+ \frac{{a}^{n}}{n!}\right) + {a}^{n+1}$

$(n+1)!\left( \sum _{ k=0 }^{ n+1 }{ \frac { { a }^{ k } }{ k! } } \right) = (n+1)\cdot {P}_{a}(n) + {a}^{n+1} = {P}_{a}(n+1)$

Notice that the sum $\sum _{ k=0 }^{ n }{ \frac { { a }^{ k } }{ k! } }$ approaches ${e}^{a}$ for large $n$.

Hence, ${ P }_{ a }\left( n \right) \sim { e }^{ a }n!$ for large $n$.

Note: the way I found this recurrence relation is by investigating the integral ${ P }_{ a }(n)=\int _{ a }^{ \infty }{ { e }^{ a-x } } { x }^{ n }dx .$ As an exercise, prove that this integral is equivalent to the defined recurrence equation.

Check out my other notes at Proof, Disproof, and Derivation