# Tagged Questions

0answers
33 views

### Generalization of small set/large set

A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges. ...
2answers
134 views

### Closed form of a sum of binomial coefficients?

I have the following function: $T_n(d)=\sum\limits_{k=\frac{n-d}{2}}^{\lceil \frac{n}{2} \rceil}{k\choose \frac{n-d}{2}}$ ${n \choose 2k}$, where $n,d\in \mathbb{N}^0$, and $n,d$ have the same ...
3answers
100 views

### Given the sequence $3, 4, 11, 16, 42\ldots$ how can I derive a general formula for it?

Given a sequence $3, 4, 11, 16, 42\ldots$ how can I derive a general formula for this sequence? Is there any optimised approach? My approach: the given series is equal to summation of $\binom{n}{k}$ ...
1answer
45 views

### Permutations through different points

I'm watching Next (2007) and I'm trying to figure out a formula. The premise of the movie is that the protagonist can look into the future for two minutes and he is able to use this to alter his ...
0answers
35 views

### Real sequence satisfies a combinatoric uniform property

Does there exist a sequence of real numbers $\{a_n\}_{n\in \mathbb{Z}_{>0}}$such that, for any fixed $k\in \mathbb{Z}_{>0}$, then $a_1, \cdots, a_k$ has a bijection to ...
1answer
22 views

### Chong inequalites about permutations

I read about two inequalities called Chong's inequalities. They state: $$\sum_{k=1}^N\dfrac{a_k}{a_{\pi(k)}}\ge N$$ and $$\displaystyle\prod_{k=1}^Na_k^{a_k}\ge\prod_{k=1}^N a_k^{a_{\pi(k)}}$$ I ...
1answer
48 views

3answers
107 views

### Binary sequence count of unique patterns

A binary sequence is a sequence of 1s and 0s, and there are $2^n$ such sequences of length $n$. Define the "pattern" as the number of consecutive $1$s in the sequence. For example, when $n=5$, the ...
1answer
95 views

### A sum for stirling numbers Pi, e.

In this identity $$1-e{}^{2} = \displaystyle \sum _{n=0}^{\infty } \frac{(-1)^n(\pi )^{2 n}} {(2 n)!}\sum _{k=0}^{2 n} (-1)^{k} S_2(2 n,1-k+2 n),$$ $S_2$ is a Stirling number of the second kind. ...
1answer
32 views

### A Sperner-like bound

Let $x_1,\cdots , x_n$ be a sequence of real number such that $x_i\geq 1$ for all $1\leq i\leq n$, $S=\{\alpha_1x_1+\cdots +\alpha_nx_n | \alpha_i\in\{0,+1,-1\}\}$ and $I=[a,b)$ be a Interval with ...
0answers
49 views

### How many “minimal sequences” are there?

A coin is tossed repeatedly and the outcome is recorded as a sequence of H's and T's. We are interested in obtaining every possible n-bit string as contiguous subsequences of our coin tossing ...
0answers
42 views

### Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
0answers
31 views

### Frogs on lotus trees [duplicate]

$n(>1)$ lotus leaves are arranged in a circle. A frog jumps from a particular leaf by the following rule: It always moves counter clockwise. From starting point it skips one leaf and jumps to the ...
0answers
32 views

### What is the number of condensed products in each term of this sequence?

Consider the sequence of polynomials defined by $$a_0 = q_0$$ $$a_{n+1} = q_{n+1} \sum_{k=0}^{n} a_k a_{n-k}$$, $q_j$ numeric variables (natural, integer, real, or complex). Each term $a_n$ is a sum ...
0answers
55 views

### Puzzle with character order

Suppose I have 3 letters a, b, c and I want to find the minimum length of a string that uses all the double combinations of the aforementioned letters. How should I do it or how are such problems ...
1answer
44 views

### $n^n$ cannot be expressed as a recurrence with polynomial coefficents

We say that a sequence $a(n)$ is $P$-recursive if there exist polynomials $p_0(n),\ldots,p_k(n) \in \mathbb{Q}[n]$ such that $$p_k(n) a(n+k) + \cdots p_0(n) a(n) = 0.$$ I would like to show that the ...
3answers
301 views

### Help with combinatorial proof of identity: $\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$

How to prove this identity? Can someone please give me some insight ? $$\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$$
0answers
33 views

### Generating function counting quaternay sequence.

I have the following problems: $1.$ Calculate the number of the n-digits Quaternary sequence containing even $"2"$ and $"1"$ and at least one $"3"$. (When a sequence is made by the digits $1,2,3,4$) ...
2answers
93 views

### why generating function $A(z) = 1 + z + z^2 + \cdots$ can be denoted as $\frac{1}{1-z}$

It is easy to see that $1 + z + z^2 + \cdots$ is equal to $\frac{1}{1-z}$ when $1 > z > 0$ and for $z >= 1$, they are not equivalent. So I have thought $\frac{1}{1-z}$ is just a short for the ...
0answers
31 views

### Binomial identities of $nx$ and $n(n-1)x^2 +nx$

I have just started learned some basic things about the binomial theorem, for fun. I have seen that $1=\sum_{r=0}^{n} {n \choose r} x^r (1-x)^{n-r}$, and I would like to use this fact to prove that: ...
0answers
23 views

### What are all the possible sums (and how often do they occur) of a k-subsequence of an n-sequence of integers?

Let $A_n = \{a_1,\dots,a_n\}$ be a sequence of non-decreasing non-negative integers. Let $P(A_n,k)$ be the set of all subsequences of $A_n$ of length $k$. Given $n,k\in\mathbb Z_{\geqslant0}$ with ...
0answers
49 views

### Approximation of the following mathematical formula

I have the following mathematical expression which I need to simplify: $$\mu^2\sum_{x=0}^{n}\left(\frac{\theta}{\mu}\right)^x\frac{1}{H_x}{n+a\choose x}$$ $\mu$, $\theta$, $D$, and $a$ are ...
0answers
38 views

### How to generalize the Thue-Morse sequence to more than two symbols?

The Thue-Morse sequence is defined as a binary sequence and can be generated like 0, 01, 01 10, 01 10 10 01, 01 10 10 01 10 01 01 10, ... . So the second half of the series is always the binary ...
1answer
21 views

### Minimum number of consecutive elements that must be chosen when choosing $\frac{3n}{4}$ elements from a sequence of length $n$

Given a sequence of length $n$, $S = (x_1 \cdot x_2 \cdot \ldots \cdot x_n$), I need to choose $\frac{3n}{4}$ elements such that I minimize the choice of consecutive elements (called a "square"). ...
0answers
34 views

### Determing sequence from its Dirichlet series

Suppose I know the Dirchlet series $$\sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \frac{\zeta(s)}{\zeta(3s)},$$ where $\zeta(s)$ is the usual Riemann zeta function. My question is - is there a way to ...
1answer
32 views

### Number of configurations in a constrained nested loops and configuration back from serial

Consider 4 counters looping the digits 0, 1, 2 to form the various "configurations", like in : ...
1answer
64 views

### Summation of product of combinations

my question is, can the following series be solved $$\sum_{i,j}^{} {a\choose i} {b \choose j}$$ where, (i+j) mod 3 =0 or i+j is multiple of 3 I need a generalized solution, i.e variables i,j,k... ...
1answer
51 views

### How many random cards picks with replacement are required?

You pick 1 card from a standard deck of 52 cards. Then put it back in, and pick a card again. Then put it back in and pick a card. etc... How many times do you have to repeat in order to have ...
1answer
38 views

### Advice on proving a tricky inequality

Im a little out of my depth here and am not well versed in combinatorics. Im not sure if this problem is too hard to solve or if there exists well known results to prove it. Here is part 1 which might ...
2answers
81 views

### Understanding the partition function

I have been trying, as a toy problem, to implement in either the Python or Haskell programming languages functions to calculate the partitions for a number and the count of those partitions. I have no ...
1answer
38 views

### A bijective transform that cycles. Help with definitions requested

In many ways I am a novice with mathematics. My background is college algebra. I am attempting to write my first maths paper and am faced with sifting through mathematics I am not familiar with. It ...
2answers
53 views

### What is the sum of $1^3q + 2^3q^3 + 3^3q^3 +\cdots+ n^3q^n$?

What is the sum of $1^3*q + 2^3*q^2 + 3^3*q^3 +...+ n^3*q^n ?$
1answer
174 views

### Number of binary n x m matrices, with at most k consecutive number of 1 in each column

I am trying to compute the number of $n x m$ binary matrices with at most $k$ consecutive values of $1$ in each column. I've figured out that I it will be enough to find the vectors with $1$ column ...
2answers
290 views

### Proof of $\sin^2(x) + \cos^2(x)=1$ using series

I have to prove the following identity $\sin^2 (x) + \cos^2(x)=1$. I can easily prove this, but this exercise is given in the section introducing the series expansions for $\sin(x)$ and $\cos(x)$ and ...
1answer
201 views

### Sequences, sets and element position in the set.

I have a sequence Q with the length of N. This is the fragment of this sequence: 68 70 72 74 76 78 80 The sequence has been divided into the sets of 4 elements ...
1answer
46 views