# 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 Note by Steven Zheng
6 years, 4 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$