I'm confused about shifting the index of function series in some situations, in particular where the expression with $n$ that is set equal to a new parameter $t$ is not a function that goes in all $\mathbb{N}$. I'll make an example.
Consider a function $f(n)$ of $n$. If I have a series like $$\sum_{n\geq0} f(n) \,\,x^{n+1}\tag{A}$$ I have no problem since I can set $n+1=t$ and then write $$\sum_{t\geq1} f(t-1) \,\,\, x^t$$ And that's should be the same exact thing as $(A)$. Here the "function" at the exponent of $x$, $g(n)=n+1$ goes from $\mathbb{N}$ in all $\mathbb{N}$.
But in some cases like this one: $$\sum_{n\geq0}f(n) \,\,\, x^{2n+1}\tag{B}$$ If I set $2n+1=t$ and then write $$\sum_{t\geq1} f(\frac{t-1}{2}) \,\,\,x^t$$ I don't think that this is the same series of $(B)$ because the parameter $t$ takes all natural numbers (even and odd), while $2n+1$ just takes odd natural numbers. In other words, the "function" in the exponent of $x$, $g(n)=2n+1$ goes from $\mathbb{N}$ in the odd natural numbers and that's a problem because the parameter $t$ is in all $\mathbb{N}$.
In general can one shift the index of a series setting $t=g(n)$ with $g(n)$ a function that does not go in all $\mathbb{N}$ but just in some subset of it?
It looks like it should not be allowed because the new series obtained "has more terms" (in the example also even numbers are considered for $t$ but not for $2n+1$).
So, if the shifting I made is incorrect, is there a way to indeed transform series like $(B)$ in power series, that is to have the exponent of $x$ (which is the "function" $g(n)$ I mentioned) as the parameter over which the sum is made?