Intereting Posts

The median minimizes the sum of absolute deviations
a question about germs of functions
Inner product of two continuous maps is continuous
Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other parabola.
Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer.
Does a finite game that cannot be drawn imply a winning strategy exists?
Bounds on $\sum_{k=0}^{m} \binom{n}{k}x^k$ and $\sum_{k=0}^{m} \binom{n}{k}x^k(1-x)^{n-k}, m<n$
Conditional probabilities from a joint density function
Game theory textbooks/lectures/etc
The cardinality of $\mathbb{R}/\mathbb Q$
Finite Field Extensions and the Sum of the Elements in Proper Subextensions (Follow-Up Question)
Prove that formula is not tautology.
n tasks assigned to n computers, what is the EX value of a computer getting 5 or more tasks?
Universal closure of a formula
functions with eventually-constant first difference

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?

- Generating function with Stirling's numbers of the second kind
- Closed formula for the sums $\sum\limits_{1 \le i_1 < i_2 < \dots < i_k \le n} i_1 i_2 \cdots i_k $?
- Roots of unity filter, identity involving $\sum_{k \ge 0} \binom{n}{3k}$
- Vandermonde-type convolution with geometric term
- About the Stirling number of the second kind
- What is the family of generating functions for the *rows* of this Stirling-number matrix for whose columns they are $\exp(\exp(x)-1)-1 $?

- Subgroup generated by $1 - \sqrt{2}$, $2 - \sqrt{3}$, $\sqrt{3} - \sqrt{2}$
- How can adding an infinite number of rationals yield an irrational number?
- Without using prime factorization, find a prime factor of $\frac{(3^{41} -1)}{2}$
- Nature of the series $\sum\limits_{n}(g_n/p_n)^\alpha$ with $(p_n)$ primes and $(g_n)$ prime gaps
- First 10-digit prime in consecutive digits of e
- A triangular representation for the divisor summatory function, $D(x)$
- Cardinality of sum-set
- Elementary proof of the Prime Number Theorem - Need?
- Sum of greatest common divisors
- Number Theory: Ramification

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.

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- Integrating squared absolute value of a complex sequence
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- Definition of limit
- Book about different kind of logic
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- Let $\frac{{{x^2}}}{{1 + {x^4}}} = \frac{1}{3}$.What is $\frac{{{x^4}}}{{1 + {x^8}}}$?
- Smallest closed ball enclosing a compact set
- Is the square root of $4$ only $+2$?
- How to properly translate the coefficients of a Taylor series?