Find the generating function for the finite sequence 0,0,0,1,2,3,4,5

Question

I have the sequence $\{0,0,0,1,2,3,4,5\}$ and I need to find a generating function.

I found the answer to be $\sum n x^{n+2}$. However to find the nth term, I would use $n-2$, correct? But then for $n=0,1,2$, I would have a negative. How do I account for those zeroes?

Answer 1

Your generating function is $x^3+2x^4+3x^5+4x^6+5x^7$

which you can write as

• $\displaystyle \sum_{n=0}^{5} n x^{n+2}$
• $\displaystyle x^2 \sum_{n=0}^{5} n x^{n}$
• $\displaystyle \sum_{n=2}^{7} (n-2) x^{n}$
• $\displaystyle \left(\sum_{n=0}^{7} (n-2) x^{n}\right) + 2 + x $
• $\dfrac{x^3(1-6x^5+5x^6)}{(1-x)^2}$ or as a variety of other things

Answer 2

You were not accurate enough by omitting the index range in the summation.

The generating function is $$0x^0+0x^1+0x^2+1x^3+2x^4+3x^5+4x^6+5x^7=\sum_{n=3}^7(n-2)x^n.$$

The zeroes are accounted for by just omitting the corresponding powers.

Note that the following representations are equivalent:

$$\sum_{n=2}^7(n-2)x^n=\sum_{n=3}^7(n-2)x^n=\sum_{n=0}^5nx^{n+2}=\sum_{n=1}^5nx^{n+2}.$$

In no case do you have to evaluate a negative coefficient.

If you really want to represent all terms, use $$\sum_{n=0}^7\max(n-2,0)x^n.$$ Another option is to indeed use negative terms, but compensate them: $$\sum_{n=0}^7(n-2)x^n+2+x.$$