Thanks, thanks, thanks!

@user180834: You’re very welcome!

You are great man! You explained me really good. :) Can I with my original proof determine function and show, that they are equal?

If you don't understand, tell me it. English isn't my native language so I may make a mistake.

@user180834: I’m glad that it helped. I’m not sure what you’re referring to when you speak of your original proof. It is possible to start with your sequence and actually derive the generating function $\frac{x^7}{1-x}+\frac{2x^7}{1-x^2}$ without knowing ahead of time, using the ideas in my answer, but you don’t have to do quite that much, since you already know what the function is.

Ok, but I would like: I assume that I don't know the generating function and I have a sequence: $0,0,0,0,0,0,0,3,1,3,1.....$ And now I detemine function generating for it and check that it is equal given. Ok?

@user180834: Yes, you can do that.

Could you give me an advice to find generating function for 0,1,2^2, 3^2 ....?

So you want the sequence $a_n=n^2$, meaning that the power series is $\sum_{n\ge 0}n^2x^n$?

By the way, to read the math symbols you want to have Chatjax, which you can find here.

Always I prefer thinking a lot on my own, but this time I have a problem. I must do some tasks to University. It's a midnight. Normally, I don't overuse your help. Forgive me ;)

I’ll get you started, at least. Let $f(x)=\frac1{1-x}$; you know that $f(x)=\sum_{n\ge 0}x^n$. Now differentiate: $f\,'(x)=\sum_{n\ge 0}nx^{n-1}$. If you multiply by $x$, you find that $$xf\,'(x)=\sum_{n\ge 0}nx^n\;.$$ Now do it again to get what you want.

Guess, from what country I come ( Europe) :D

Europe, and it’s midnight?!

ok, It's near 3 o'clock

and I don't go to sleep today :)

That’s tough: that time zone covers most of central and eastern Europe.

so, there is a high probability that you guess :)

You have a little trouble using the articles (the, a) correctly, so my first guess would be a Slavic-speaking country.

yes

Still too many possibilities; Poland?

thanks for an advice. It helped.

yes

You’re welcome; I’m glad that it was helpful.

Can I have another question?

I promise, last :)