Marcos

Posted on Wednesday, 04 June, 2003 - 08:14 pm:

I was recently looking at a P2 question that involved working out

(1 + y + y^{2})^{3}

or something, and because I got bored I decided to expand other powers of this and see what I get. I noticed a very interesting (at least, I think so) pattern. If you write the coefficients in a triangular array similar to Pascal's triangle, you get the same principle but instead of adding the two numbers above you add the three numbers above.

I've tried this up to $(1 + x + x^{2})^{5}$ and it definitely works but it's too boring to do any more...

Basically, I'm asking if this is true, and if so:

Question 1

What is the formula for the general coefficient? (Once I've got this I guess I can prove the expansion by induction.)

If we let $μ_{r} (n)$ be the coefficient of $x^{r}$ in $(1 + x + x^{2})^{n}$ , I've worked out that:

$μ_{0} (n) = 1$

$μ_{1} (n) = n$

$μ_{2} (n) = n (n + 1) / 2$

$μ_{3} (n) = n (n - 1) (n + 4) / 6$

$μ_{r} (n) = μ_{2 n - r} (n)$

I can't see a general formula for $μ_{r} (n)$ though...

Question 2

Can we extend this to the expansion of $[\sum_{r = 0}^{t} x^{r}]^{n}$ ? By this I mean the 'Pascal's triangle'-like thing aswell as the general coefficient of $x^{i}$ ?

PleaseNote:Before anyone starts charging away giving full and clear answers, all I want is basically a few hints just to see that I'm on the right track and how I should proceed in finding a general formula for the coefficients. (Combinatorics-related hints are also more than welcome!)

Thanks,

Marcos

David Loeffler

Posted on Wednesday, 04 June, 2003 - 08:21 pm:

I'm afraid you're on a hiding to nothing there, Marcos. Take a look at MathWorld 's entry on the subject; if you follow up the references (Zeilberger's book is available for free online, and is well worth reading) you'll see that it has been proved that there is no nice formula in terms of factorials or anything like that for the central coefficient (the coeff of xⁿ ). So there is certainly no general formula for the (n,k)^th coefficient.

David

Marcos

Posted on Wednesday, 04 June, 2003 - 08:30 pm:

Yikes!

Thanks David, I guess I'd better stop trying to work out a general formula then. I was trying to construct a function which is at least close to the rth derivative of $(1 + x + x^{2})^{n}$ with no luck. Now I can see why...

How about my second question?

For [ $\sum_{r = 0}^{t} x^{r}$ ] will the coefficients always form a Pascal's-triangle-like array, where one number is the sum of the $t + 1$ above it?

Marcos

David Loeffler

Posted on Wednesday, 04 June, 2003 - 08:39 pm:

Yes