All Questions
21
questions
6
votes
1
answer
228
views
An alternating sum
I ran into an alternating sum in my research and would like to know if the following identity is true:
$$
\sum_{i = 0}^{\left\lfloor \left(n + 1\right)/2\right\rfloor} \frac{\left(n + 1 - 2i\right)^{n ...
- 147
2
votes
1
answer
202
views
If $xy = ax + by$, prove the following: $x^ny^n = \sum_{k=1}^{n} {{2n-1-k} \choose {n-1}}(a^nb^{n-k}x^k + a^{n-k}b^ny^k),n>0$
If $xy = ax + by$, prove the following: $$x^ny^n = \sum_{k=1}^{n} {{2n-1-k} \choose {n-1}}(a^nb^{n-k}x^k + a^{n-k}b^ny^k) = S_n$$ for all $n>0$
We'll use induction on $n$ to prove this.
My ...
1
vote
1
answer
35
views
Difference of two powers where a term is multiplied by some constant
We know that the difference of two powers $x^n-y^n$ over its linear factor $x-y$ is
$$\frac{x^n-y^n}{x-y}=\sum_{j=0}^{n-1}(x^{n-j-1}y^{j}),$$
where $n\in\mathbb{N}$ and all $x,y\in\mathbb{F}, \ \...
- 81
2
votes
4
answers
62
views
For a fixed $k$ what is the value of $\sum_{l=1}^{5^m-1} \Big\lfloor \dfrac{l}{5^k}\Big \rfloor$
For a fixed $k$ what is the value of $\sum_{l=1}^{5^m-1} \Big\lfloor \dfrac{l}{5^k}\Big \rfloor$
By dividing the numbers between $1$ and $5^m$ as intervals of $5^k$, I was getting the following ...
user675768
2
votes
1
answer
63
views
For any prime $p>3$ show that $C_{np}^{p}-C_{np}^{2p}+C_{np}^{3p}-C_{np}^{4p}+...+(-1)^{n-1}C_{np}^{np} \equiv 1\pmod{p^3}$
Let $n$ be a positive integer. For any prime $p>3$ show that
$$C_{np}^{p}-C_{np}^{2p}+C_{np}^{3p}-C_{np}^{4p}+...+(-1)^{n-1}C_{np}^{np} \equiv 1\pmod {p^3}$$ Where $C_{n}^{k}=\frac{n!}{k!(n-k)!}$. (...
- 11.6k
0
votes
1
answer
180
views
How to prove $f(n)=\sum_{k=0}^{n}C_{2n+k}^{n}C_{n-1+k}^{n-1}\equiv1\pmod2$?
Prove that $$f(n)=\sum_{k=0}^{n}\binom{2n+k}{n}\binom{n-1+k}{n-1}\equiv1\pmod2,\quad\forall n\in N^{+}$$
I have tried:
By Lucas TH, we have $f(2n)\equiv f(n)\pmod 2$, but $$f(2n+1)=\sum_{k=0}^{n}\...
- 93.6k
4
votes
0
answers
146
views
A $p$-congruence involving binomial coefficients and the floor function.
Let $p$ be a prime number.
Let $k$ and $m$ be positive integers such that $k \ge m$ and $p-1$ does not divide $k-m$.
It seems to be true that $$ (-1)^{m-\lfloor
\frac{k-m}{p-1}\rfloor}\sum_{i\...
- 3,716
2
votes
2
answers
326
views
Compute telescopic sum of binomial coefficients
Is there a nice or simple form for a sum of the following form?
$$ 1 + \sum_{i=1}^k \binom{n-1+i}{i} - \binom{n-1+i}{i-1}$$
Motivation: Due to a computation in the formalism of Schubert calculus the ...
- 448
4
votes
1
answer
260
views
Role of binomial coefficents in nested summations in layman terms
I has this doubt from almost two years and not getting a simple solution in layman terms. In short, the doubt is the about relation between binomial coefficients and the nesting summation.
Recently I ...
- 196
1
vote
0
answers
109
views
Is it true that $ \sum_{k=0}^{(2q+1)n+q} \binom{2\left((2q+1)n+q\right)+1}{2k} 3^k $ is divisible by $\sum_{k=0}^{q} \binom{2q+1}{2k} 3^k $
This is a continuation for this question.
Let $$S_q=\sum_{k=0}^{q} \binom{2q+1}{2k} 3^k $$
$$T_q=\sum_{k=0}^{q} \binom{2q+1}{2k+1} 3^k $$.
(i) Is it true that $S_{(2q+1)n+q}$ is divisible by $S_q$, ...
- 3,716
0
votes
3
answers
2k
views
How to prove the sum of triangular numbers in the form n(n+1)(n+2)/6? [duplicate]
Show that the sum of the first $n$ triangular numbers is
$n$($n + 1$)($n + 2$)/6
- 9
4
votes
3
answers
243
views
Prove $z^n + 1/{z^n}$ is rational if $z+1/z$ is rational
Knowing that $z + \frac{1}{z} = 3$ ($z \in \mathbb{R}$), prove that $z^n + \frac{1}{z^n} \in \mathbb{Q}$ ($n \in \mathbb{N}$ and $n \geq 1$).
I've tried to find a general formula for $z^n + \frac{1}{...
- 2,833
2
votes
2
answers
371
views
Finite summation including binomial coefficients and double factorials
I came across the following summation:
$$
\sum_{k=0}^n\frac{(-1)^k(2k)!!}{(2k+1)!!}\dbinom{n}{k}\,\,\,\,(n\in\mathbb{N}).
$$
$\tbinom{n}{k}$ are binomial coefficients, $n!/k!(n-k)!$.
Mathematica told ...
- 35
-2
votes
1
answer
68
views
Recovering the coefficients $b_r$ of the binomial sum $\sum_{r=0}^n\binom{n}rb_r$ [closed]
Suppose that the sequences of real numbers $a_0,a_1,a_2,a_3,\ldots$ and $b_0,b_1,b_2,b_3,\ldots$ satisfy the relation $$a_n=\sum_{r=0}^n\binom{n}rb_r\;.$$ Then prove that $$(-1)^nb_n=\sum_{s=0}^n\...
- 31
1
vote
2
answers
351
views
Inductive proof of the identity $\binom{n} {0} F_0+\binom{n}{1} F_1+\binom{n}{2} F_2+\cdots +\binom{n}{n} F_n=F_{2n}$ [duplicate]
I'm trying to prove the following identity:
$$\binom{n} {0} F_0+\binom{n}{1} F_1+\binom{n}{2} F_2+\cdots +\binom{n}{n} F_n=F_{2n}$$
I need to prove it using induction (not a counting argument), I ...
