4
$\begingroup$

Can anyone tell me if there is a formula for finding the coefficient of $x^3$ in the expansion of $(3x+5)_{6}$, where $(a)_n$ denotes the Pochhammer symbol, i.e. $(a)_{n}=a\cdot(a+1)\cdots(a+n-1)$?

It would be even better if someone could explain how I might derive a general formula for the coefficient of $x^k$ in the expansion $(ax+b)_n$.

$\endgroup$
3
$\begingroup$

The coefficient of $x^k$ in $(x)_n$ is the signless Stirling number of the first kind, that is $c(n,k)$ the number of permutations of $n$ elements with $k$ cycles.

A lot is known about these numbers, recurrence formulas, sum formulas, generating functions, etc, and it is easy to see that this coefficient has the same recurrence formula, see http://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind

Now, you are looking for the coefficient of $x^k$ in $(ax+b)_n$, which gives you $$c(n,k) a^k b^{n-k}$$ because you pick up a factor $a$ for every $x$ and you pick up a factor $b$ for every parenthesis where you do not choose $x$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.