Intereting Posts

Given a point $x$ and a closed subspace $Y$ of a normed space, must the distance from $x$ to $Y$ be achieved by some $y\in Y$?
Number of normal subgroups of a non abelian group of order $21$… CSIR December $2013$
Car movement – differential geometry interpretation
The Cantor ternary set is totally disconnected
Deriving the analytical properties of the logarithm from an algebraic definition.
Proving that $S_n$ has order $n!$
$\ell^1$ vs. continuous dual of $\ell^{\infty}$ in ZF+AD
Uniqueness of Duals in a Monoidal Category
Prove any function $f$ is Riemann integrable if it is bounded and continuous except finite number of points
Prove that if $\ker(T) \subseteq \ker(S)$, then $S = kT$ for some $k\in \mathbb{R}$
Simplification of expressions containing radicals
Interchanging the integral and the infinite sum
Do we know if there exist true mathematical statements that can not be proven?
Why is there antagonism towards extended real numbers?
Are isometries always linear?

For example if I have the generating function $\frac{1}{1-2x}$ then it corresponds to the sequence $1 + 2x + 4x^2 + 8x^3 +~…$. I know how to start from the sequence and get the generating function, but I don’t know how to start from the generating function and get the sequence.

Similarly, what if I have a generating function like $\frac{1}{1+x}$? How do I get the corresponding sequence? Same way?

- Closed form of an inhomogeneous non-constant recurrence relation
- The generating function for permutations indexed by number of inversions
- Find a generating function for the number of strings
- Generating functions - combinatorics problem of Mirko couting vehicles through the toll
- Roots of unity filter, identity involving $\sum_{k \ge 0} \binom{n}{3k}$
- Generating function of $a_{n}^2$ in terms of GF of $a_{n}$?

- Convergence of the sum of two infinite series only at $x=\frac12$?
- The generating function for the Fibonacci numbers
- Primes for >1 (good expression)
- $x^2+x+1$ is the cube of a prime.
- $5$-adic expansion of $−2$
- Prove that the sum of digits of $(999…9)^{3}$ (cube of integer with $n$ digits $9$) is $18n$
- Permutation Partition Counting
- Integral solutions of $x^2+y^2+1=z^2$
- How to prove that if $a\equiv b \pmod{2n}$ then $a^2\equiv b^2 \pmod{2^2n}$
- When is $\sin x$ an algebraic number and when is it non-algebraic?

Maybe the following may help you. If you have a linear recurrence of the form

$$

u_{n+2}+a\cdot u_{n+1}+b\cdot u_n=0 \tag1

$$ then, multiplying out $(1)$ by $x^{n+2}$ and summing one gets

$$

\sum_{n=0}^\infty u_{n+2}x^{n+2}+a\cdot \sum_{n=0}^\infty u_{n+1}x^{n+2}+b\cdot \sum_{n=0}^\infty u_nx^{n+2}=0\tag2

$$ or, with changes of index,

$$

\sum_{n=2}^\infty u_{n}x^n+a\cdot x\sum_{n=1}^\infty u_{n}x^{n}+b\cdot x^2\sum_{n=0}^\infty u_nx^{n}=0\tag3

$$

$$

\sum_{n=0}^\infty u_{n}x^n+a\cdot x\sum_{n=0}^\infty u_{n}x^{n}+b\cdot x^2\sum_{n=0}^\infty u_nx^{n}=u_0+(u_1+au_0)\cdot x\tag4

$$ that is, by factorizing $\displaystyle \sum_{n=0}^\infty u_{n}x^n=f(x)$,

$$

(1+ax+b x^2)f(x)=u_0+(u_1+au_0)\cdot x

$$ and

$$

f(x)=\frac{u_0+(u_1+au_0)\cdot x}{1+ax+b x^2} \tag5

$$

Going from $(1)$ to $(5)$ and vice-versa is a usual link between a generating function and a recurrent relation between its coefficients.

- How to find sums like $\sum_{k=0}^{39} \binom{200}{5k}$
- For every continuous function $f:\to$ there exists $y\in $ such that $f(y)=y$
- What is the minimum polynomial of $x = \sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6} = \cot (7.5^\circ)$?
- Maximum of Polynomials in the Unit Circle
- knowledge needed to understand Fermat's last theorem proof
- Non-orthogonal projections summing to 1 in infinite-dimensional space
- Find the transitional matrix that would transform this form to a diagonal form.
- $K$ is a normal subgroup of a finite group $G$ and $S$ is a Sylow $p$-subgroup of $G$. Prove that $K \cap S$ is a Sylow $p$-subgroup of $K$.
- If $a_n$ goes to zero, can we find signs $s_n$ such that $\sum s_n a_n$ converges?
- A question about the arctangent addition formula.
- functions with eventually-constant first difference
- universal property in quotient topology
- joint probability distribution of one discrete, one continuous random variable
- The equation $x^3 + y^3 = z^3$ has no integer solutions – A short proof
- Solving a double integration in parametric form