0
votes
0answers
16 views

Dependent Expectation in Random Numbers Illustrated by Prime Repetition in Pi

When approximating Pi, appending each numerical digit as you refine, what is the first repetition of a four-digit prime number? For instance the first repetition of any one-digit number in the ...
3
votes
1answer
54 views

An equality from Representation Theory

Studying Representation Theory of finite groups I've bumped in the following identity: $$\frac{n(n+1)}{2}=\sum_{i=1}^n\frac{(2i-1)!!(2n-2i+1)!!}{(2i-2)!!(2n-2i)!!}$$ My book suggests to prove it ...
0
votes
0answers
22 views

Unique sum between elements of a numerical set

I require a set of numerical elements on wich the sum of some of these elements is unique to the set, that it's to say, no other combination in the sum of elements will result the same outcome. ...
0
votes
0answers
25 views

Given a positive integer $k$, find the integer part of $n^2 /k$ for $n\ge 1$, and a related question.

For a given positive integer $k,$ I am looking for possible answers / literature about the sequence $(a_n)=([\frac{n^2}{k}])_{n=1}^\infty$, where $[x]=$the integer part of $x.$ This question is ...
1
vote
2answers
44 views

Closed form sum for the series given below?

Does the following series have a closed form sum? $$f(n,r) = \sum_{i=0}^n \binom{r+i}{r}$$
1
vote
3answers
39 views

Why is $\sum\limits_{i=0}^{n}(r-1)^{n-i}{n\choose i} = r^n$?

I was solving a problem and found that $\sum\limits_{i=0}^{n}2^{n-i}{n\choose i} = 3^n$. So I tried to generalise it and got $\sum\limits_{i=0}^{n}(r-1)^{n-i}{n\choose i} = r^n$. Is it true for $r ...
2
votes
0answers
25 views

Evaluate $ \sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ where $n,k$ are fixed

Is there a general way/technique to evaluate $ \sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ in terms of $r$, where we consider $n$ and $k$ fixed natural numbers and $n > k$? (here, ...
0
votes
1answer
290 views

provide a combinatorial proof that $C_{n+1} = C_0C_n + C_1C_{n-1} + …. + C_kC_{n-k} + …C_nC_0$

(a) Let $C_n$ denote the number of ways of writing a valid list of open and closed parentheses of length $2n$ (valid means that at any point along the list, the number of open parentheses must be ...
4
votes
5answers
124 views

Sums $\sum_{k = 0}^n k^t {n \choose k}$ where $t$ is a positive integer

I recently came across the problem of finding out the sum $\sum_{k = 0}^n k^2 {n \choose k}$. The solution that I've found goes something like this: $\sum_{k = 0}^n k^2 {n \choose k}=\sum_{k = 0}^n ...
0
votes
1answer
46 views

How to evaluate $\sum_{k=0}^{n} \alpha^k \binom{n}{k}$?

I am trying to show that the function that satisfies $f^\prime(x)=f(x)$ with $f(0)=1$ behaves in an exponential way (in other words, I want to justify writing it as $e^x$). I need to show that: $$ ...
3
votes
0answers
34 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. ...
4
votes
3answers
163 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 ...
0
votes
3answers
107 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 $\binom{n}{k}$ ...
1
vote
1answer
46 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 ...
0
votes
0answers
36 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 ...
0
votes
1answer
26 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 ...
1
vote
1answer
49 views

Infinite series for recurrence

Question 1 If I define $A(z) = \sum_{n \ge 0} a_n \frac{z^n}{n!} \tag 1$ (where $a_n$ are $3\times 3$ constant matrices indexed with n), then can we re-write $\sum_{n \ge 1} a_{n-1} \frac{z^n}{n!} ...
4
votes
1answer
44 views

What is this sequence of all permutations with gaps permissible

Let there be a sequence $a_1, a_2, a_3,...,a_n$ that represent some actions that you know are required to solve a problem. However, you do not know what order these actions need to be taken to solve ...
3
votes
2answers
54 views

Proof $e^n*n!$ is an asymptote of $(n+1)^n$

I would like to prove $\lim_{n\to \infty}e^nn!-(n+1)^n=0$. All I have really done is show $(n+1)^n=\sum_{i=0}^n\frac{n!}{(n+1)^i(i!)(n-i)!}$
1
vote
1answer
39 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
1
vote
1answer
44 views

Rational Series VS Algebraic Series

I am reading a paper on combinatorics. It mentions some generating functions are rational series and others are algebraic series. I do not understand the difference, can someone help? EDIT $1$: The ...
1
vote
2answers
75 views

Find $R$ such that $\sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(R)}^n\cdot{(1-R)}^{3k-n}$ is constant for all $k\in\mathbb{N}$

Given $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(R)}^n\cdot{(1-R)}^{3k-n}$ with $0<R<1$. The sequence $A_k$ seems to be decreasing for $R\leq0.6$ and increasing for $R\geq0.8$. How can ...
3
votes
2answers
89 views

Prove by induction that $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ is decreasing

I want to prove that the following sequence is monotonously decreasing: $A_k = \sum\limits_{n=2k}^{3k}\binom{3k}{n}\cdot{(\frac{60}{100})}^n\cdot{(\frac{40}{100})}^{3k-n}$ I think it should be ...
1
vote
1answer
79 views

Expected value over many trials

I am a poker player and was talking to my friend about expected value. He claimed that if you play far enough above your bankroll, expected value can be negative, even if you have a skill edge. I ...
5
votes
1answer
50 views

Closed form for sequence A145271

I would like to know if there is a simple formula or method of expanding the expression given by $\left[g(x) \frac{d}{dx}\right]^n g(x)$ where $n$ is a positive integer, without having to resort to ...
1
vote
1answer
36 views

Iterate through n coins flipping these obtaining all possible combinations.

If I have let say n coins all facing the same way. Is there an iterative method for turning these coins, one at a time, until all possible combinations have occurred one and only one time? This is ...
2
votes
0answers
26 views

An interesting identity involving triangle numbers.

Let $T^{(2)}_i$ be 1 for $i = 1,2,\dots$ and $T^{(3)}_i$ be the natural numbers $T^{(4)}_i$ be the triangular numbers $T^{(5)}_i$ be the tetrahedral numbers and so on for $i = 1,2,\dots$ For $m = ...
3
votes
3answers
116 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 ...
1
vote
1answer
96 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. ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 $k=m-1$.
2
votes
0answers
34 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 ...
2
votes
0answers
57 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 ...
2
votes
1answer
45 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 ...
10
votes
3answers
339 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 ? $$\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$$
0
votes
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 $1,2,3,4$) ...
3
votes
2answers
94 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 ...
1
vote
0answers
32 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: ...
2
votes
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 ...
1
vote
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 ...
2
votes
0answers
43 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 ...
0
votes
1answer
22 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"). ...
1
vote
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 ...
0
votes
1answer
33 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 : ...
0
votes
1answer
76 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... ...
1
vote
1answer
52 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 ...
1
vote
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 ...
1
vote
2answers
84 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 ...
0
votes
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 ...