Intereting Posts

Deriving the Normalization formula for Associated Legendre functions: Stage $3$ of $4$
Why is the range of inverse trigonometric functions defined in this way?
If we define $\sin x$ as series, how can we obtain the geometric meaning of $\sin x$?
$\lim_{\lambda \to \infty} \int^b_0 f(t) \frac{\sin(\lambda t)}{t} $
Proof that Right hand and Left hand derivatives always exist for convex functions.
Eigen Values Proof
Example of a group
Proving the summation formula using induction: $\sum_{k=1}^n \frac{1}{k(k+1)} = 1-\frac{1}{n+1}$
Finding the inverse of the arc length function
Some questions about mathematical content of Gödel's Completeness Theorem
Principal value of the singular integral $\int_0^\pi \frac{\cos nt}{\cos t – \cos A} dt$
Concrete description of (co)limits in elementary toposes via internal language?
Decomposable elements of $\Lambda^k(V)$
Necessary and sufficient condition that a localization of an integral domain is integrally closed
Formal definition of a random variable

I need to evaluate the following sum, which depends on $n \in \mathbb N$ (call it $S(n)$ if you will)

$$ \sum_{i=0}^{n} (-1)^{n-i} \binom{n}{i} f(i)$$

where $f$ defined over $\mathbb N$ is determined by the identity

- Sum $\displaystyle \sum_{n=i}^{\infty} {2n \choose n-i}^{-1}$
- A double sum with combinatorial factors
- Sum involving the product of binomial coefficients
- Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity)
- Evaluate $\sum_{k = 0}^{n} {n\choose k} k^m$
- How can I prove this combinatorial identity $\sum_{j=0}^n j\binom{2n}{n+j}\binom{m+j-1}{2m-1}=m\cdot4^{n-m}\cdot\binom{n}{m}$?

$$ \sum_{n \geq 0} f(n) \frac{x^n}{n!} = exp \left ( x+\frac{x^2}{2} \right)$$

This is a problem left as an exercise to the reader in Richard Stanley’s “Enumerative Combinatorics”, in the first few pages of Chapter 1, and I assume it should be simple but none of my approaches, including searching for identities involving binomial coefficients, have worked.

Thank you in advance!

- How many solutions possible for the equation $x_1+x_2+x_3+x_4+x_5=55$ if
- Numbering edges of a cube from 1 to 12 such that sum of edges on any face is equal
- How many ways can 70 planes be allocated into 4 runways?
- Sum of combinations of the n by consecutive k
- Combinatorial proof of $\sum_{k=0}^{n} \binom{n+k-1}{k} = \binom{2n}{n}$
- A mouse leaping along the square tile
- Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
- Number of relations that are both symmetric and reflexive
- Prove the identity Binomial Series
- What is the number of distinct 3 letter words out of different number of given letters?

**Hint:** $$\left(\displaystyle \sum_{l \geq 0} f(l) \frac{x^l}{l!}\right).\left(\sum_{m \geq 0} (-1)^m \frac{x^m}{m!} \right) = \quad …….$$

Setting

$$

S_n=\sum_{i=0}^n(-1)^{n-i}{n\choose i}f(i),

$$

we have

\begin{eqnarray}

\exp\left(\frac{x^2}{2}\right)&=&\exp\left(x+\frac{x^2}{2}\right)\exp(-x)

=

\left(\sum_{n=0}^\infty\frac{f(n)}{n!}x^n\right)\left(\sum_{n=0}^\infty\frac{(-1)^n}{n!}x^n\right)\\

&=&\sum_{n=0}^\infty\left(\sum_{i=0}^n\frac{f(i)}{i!}\cdot\frac{(-1)^{n-i}}{(n-i)!}\right)x^n

=\sum_{n=0}^\infty\left(\frac{1}{n!}\sum_{i=0}^n(-1)^{n-i}{n\choose i}f(i)\right)x^n\\

&=&\sum_{n=0}^\infty\frac{S_n}{n!}x^n,

\end{eqnarray}

i.e.

$$

\sum_{n=0}^\infty\frac{S_n}{n!}x^n=\exp\left(\frac{x^2}{2}\right)=\sum_{n=0}^\infty\frac{1}{n!}\left(\frac{x^2}{2}\right)^n=\sum_{n=0}^\infty\frac{x^{2n}}{2^nn!}.

$$

It follows that

$$

S_{2n+1}=0,\ S_{2n}=\frac{(2n)!}{2^nn!} \quad \forall n\ge 0.

$$

Here is your $f(n)$

$$ f(n) = n!\sum_{k=0}^{\lfloor n/2 \rfloor}\frac{2^{-k}}{ (n-2k)!\,k! }. $$

Related problems.

- Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations?
- Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$
- $\prod_{k=1}^\infty \cos(x2^{-k})$
- Why isn't there a good product formula for antiderivatives?
- Relation between Borelâ€“Cantelli lemmas and Kolmogorov's zero-one law
- Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?
- Does linear ordering need the Axiom of Choice?
- Is there a generalization of the fundamental theorem of algebra for power series?
- Subgroups as isotropy subgroups and regular orbits on tuples
- How to find $E(X_1X_2X_3)$ given the joint CDF?
- What is the precise definition of 'between'?
- Galois group of $\mathbb Q(\sqrt{4+\sqrt 7})/\mathbb Q$
- Cardinality of the set of all real functions which have a countable set of discontinuities
- Modular Arithmetic with Powers and Large Numbers
- Compatibility of topologies and metrics on the Hilbert cube