Please help to find term's coefficient in the following example

Question

I trying find the number of all solutions in the following:

$ x_1 + x_2 + x_3 + x_4 + x_5 = 24 $

where:

2 of variables are natural odd numbers

3 of variables are natural even numbers

none of variables are equal to $0$ or $1$ (all the variables are $>= 2$)

So I've made generating functions according to the limitations as follow:

$ (x^3 + x^5 + x^7 + ... + x^{21} + x^{23})^2 $ $ (x^2 + x^4 + x^6 + ... + x^{22} + x^{24})^3 $

Now I need to find the coefficient of $ x^{24} $

I've tried to solve it but get stuck, here is what I've done:

$ x^6(1 + x^2 + x^4 + ... + x^{18} + x^{20})^2 $ $ x^6(1 + x^2 + x^4 + ... + x^{20} + x^{22})^3 $

$ x^{12}(1 + x^2 + x^4 + ... + x^{18} + x^{20})^2(1 + x^2 + x^4 + ... + x^{20} + x^{22})^3 $

Now we actually looking for coefficient of $ x^{24-12}=x^{12} $ in:

$ (1 + x^2 + x^4 + ... + x^{18} + x^{20})^2(1 + x^2 + x^4 + ... + x^{20} + x^{22})^3 $

$ (\frac{1 - x^{21}}{1 - x})^2(\frac{1 - x^{23}}{1 - x})^3 $

Here I'm stuck. How to proceed in order to find coefficient of $ x^{12} $?

Regards.

PS: Solution was given but just out of curiosity, how to solve it using generating-functions?

Answer 1

Let the numbers be $2a + 1, 2b + 1, 2c, 2d, 2e$, where $a, b, c, d, e$ are natural. Note that this makes the odd numbers greater than or equal to $3$, and the even numbers greater than or equal to $2$.

Then you have $2a + 1 + 2b + 1 + 2c + 2d + 2e = 24 \rightarrow a + b + c + d + e = 11$. You can solve this with a standard stars-and-bars approach.

Answer 2

Let's take it from the point where we ask for the coefficient of $x^{12}$ in:

$$ (1 + x^2 + x^4 + ... + x^{18} + x^{20})^2(1 + x^2 + x^4 + ... + x^{20} + x^{22})^3 $$

[Note, however, that a factor of $\binom{5}{2}=10$ is already left out of the calculation at this point, which would account for varying positions of the two odd summands $x_i$. See the comments on Ashtay's Answer for more details.]

Replace $x^2$ with $y$ and ask for the coefficient of $y^6$:

$$ (1 + y + y^2 + \ldots + y^9 + y^{10})^2 (1 + y + y^2 + \ldots + y^{10} + y^{11})^3 $$

Of course we can remove all polynomial in $y$ terms with exponents greater than $6$ since these cannot contribute to the coefficient of $y^6$, then notice the polynomial factors are now equal:

$$ (1 + y + y^2 + \ldots + y^6)^5 $$

We can continue to explicitly work out the coefficient of $y^6$ by repeated squarings (and removing terms beyond $y^6$):

$$ (1 + y + \ldots + y^6)^2 \; \to \; (1 + 2y + 3y^2 + 4y^3 + 5y^4 + 6y^5 + 7y^6) $$

$$ (1 + 2y + 3y^2 + \ldots + 7y^6)^2 \; \to \; (1 + 4y + 10y^2 + 20y^3 + 35y^4 + 56y^5 + 84y^6) $$

Multiplication by $(1+y+y^2+\ldots + y^6)$ of the last result gives the desired term $210y^6$. It follows that $210$ was the original coefficient of $x^{24}$ you wanted (if my arithmetic is right!).