Questions tagged [combinatorial-identities]
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26 questions with no upvoted or accepted answers
46
votes
0
answers
3k
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A = B (but not quite); 3-d arrays with multiple recurrences
Many years ago, I discovered the remarkable array (apparently originally discovered by Ramanujan)
1
1 3
2 10 15
6 40 105 105
24 196 700 1260 945
...
- 6,342
18
votes
0
answers
755
views
Two curious series for $1/\pi$
On Jan. 18, 2012 I conjectured that for any prime $p>3$ we have
$$R_p^2\equiv\frac1{10}\left(512\left(\frac{10}p\right)-27\left(\frac{-15}p\right)-475\right)\pmod p,$$
where $(\frac{\cdot}p)$ ...
- 15.6k
12
votes
0
answers
629
views
$q$-analogue of the multinomial theorem?
The $q$-binomial theorem states that
$$
\prod_{k=0}^{n-1}(1+q^kt) = \sum_{k=0}^n q^{\binom k2}{n\brack k}_q t^k.
$$
This identity is a $q$-analogue of the binomial theorem
$$
(1+t)^n = \sum_{k=0}^n \...
- 5,654
10
votes
0
answers
509
views
New series for $\zeta(5)$ involving second-order harmonic numbers
In 1997 T. Amdeberhan and D. Zeilberger proved that
$$\sum_{k=1}^\infty\frac{(-1)^k(205k^2-160k+32)}{k^5\binom{2k}k^5}=-2\zeta(3).\tag{1}$$
In 2008 J. Guillera obtained that
$$\sum_{k=1}^\infty\frac{(...
- 15.6k
10
votes
0
answers
349
views
A bijective proof for the odd companion to Shapiro's Catalan convolution
Shapiro's Catalan convolution is the following formula (where $C_n$ is the $n$th Catalan number):
$$
\sum_{k=0}^{n}{C_{2k}C_{2(n-k)}}=4^nC_n.
$$
In other words, letting $C(z)=\sum_{n=0}^{\infty}{C_nz^...
- 1,190
9
votes
0
answers
192
views
For $q$-analogues of a known curious identity
In 2002 I published the folllowing curious combinatorial identity:
$$(x+m+1)\sum_{i=0}^m(-1)^i\binom{x+y+i}{m-i}\binom{y+2i}i-\sum_{i=0}^m\binom{x+i}{m-i}(-4)^i=(x-m)\binom xm.$$
My original proof is ...
- 15.6k
8
votes
0
answers
351
views
A hypergeometric series for $\sqrt3\pi$ with converging rate $1/9$
Recently, I found a (conjectural) new series for $\sqrt3\pi$:
$$\sum_{k=1}^\infty\frac{(8k-3)\binom{4k}{2k}}{k(4k-1)9^k\binom{2k}k^2}=\frac{\sqrt3\pi}{18}.\label{1}\tag{1}$$
The series converges fast ...
- 15.6k
6
votes
0
answers
235
views
A curious series for $L(2,(\frac{-3}{\cdot}))$
Let
$$K:=L\left(2,\left(\frac{-3}{\cdot}\right)\right)=\sum_{k=1}^\infty\frac{(\frac k3)}{k^2}=\sum_{j=0}^\infty\left(\frac1{(3j+1)^2}-\frac1{(3j+2)^2}\right),$$ where $(\frac k3)$ is the Legendre ...
- 15.6k
6
votes
0
answers
298
views
A new series for $\sqrt3/\pi$?
Recently, I conjectured the following identity:
$$\sum_{k=0}^\infty\frac{(66k^2+37k+4)\binom{2k}k\binom{3k}k\binom{4k}{2k}}{(2k+1)729^k}=\frac{27\sqrt3}{2\pi}.\tag{1}$$
This can be easily checked ...
- 15.6k
5
votes
0
answers
350
views
Sum over permutations involving sign
The problem is to evaluate the following sum over all permutations $\sigma\in S_{d}$ of $\{1,2,...,d\}$:
$\displaystyle\sum_{\sigma\in S_{d}}\text{sgn}(\sigma)\displaystyle\frac{1}{\prod_{i=1}^{d}(\...
- 1,538
5
votes
0
answers
345
views
When does a triangle of numbers have a zero row sum?
Suppose we have a triangle of numbers defined by the recurrence relation
$$\left| n \atop k \right| = f(n,k) \left| n-1 \atop k \right| +g(n,k) \left| n-1 \atop k-1 \right| + [n=k=0],$$
for some ...
- 3,283
4
votes
0
answers
97
views
"Convolving" a general Catalan with classical Catalan
Consider what is sometimes known as generalized Catalan sequence
$$\mathcal{{\color{red}C}}_{a,b}:=\frac{2b+1}{a+b+1}\binom{2a}{a+b}.$$
Observe that $\mathcal{{\color{red}C}}_{n,0}$ reduces to the ...
- 43.2k
4
votes
0
answers
208
views
Extract this constant term
Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term.
For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
- 43.2k
4
votes
0
answers
233
views
Curious double sum identity
The following identity seems to hold for $a>1$ :
$$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{ \frac{a^m}{m}\left( \frac{a^m}{m}+ \frac{a^n}{n} \right) } = \frac{a^2}{2(a-1)^4}$$
I've tested ...
- 5,940
4
votes
0
answers
225
views
For a combinatorial proof of a symmetric identity
In my paper Supercongruences involving dual sequences [Finite Fields Appl. 46(2017), 179-216], I gave a new symmetric identity which states that if $x+y=-1$ then
$$\sum_{k=0}^n(-1)^k\binom xk^2\binom{...
- 15.6k
4
votes
0
answers
312
views
Generalization of Tamarkin’s ARO 1993, final round, problem 10/8: part II
Let us use the notations of my previous question about Tamarkin's problem.
Let $\ell\in\left\lbrace 0,1,...,p\right\rbrace$.
An element $f\in \mathbb Z^{\mathbb Z}$ is said to be $\ell$-average-...
- 33.8k
3
votes
0
answers
172
views
Equality of determinants: a direct justification request
Let $I_k$ denote the enumeration of involutions among permutations in $\mathfrak{S}_k$. Here is yet another cute finding for which I ask a:
Question. Is there a direct proof (or interpretation or ...
- 43.2k
2
votes
0
answers
196
views
For human proofs of two novel combinatorial identities
For $n=0,1,2,\ldots$, let us define the polynomial
$$S_n(x):=\sum_{k=0}^n\binom{x/2}k\binom{(x-1)/2}k\binom{-(x+1)/2}{n-k}\binom{-(x+2)/2}{n-k}.$$
Such polynomials occur in some series for $1/\pi$ ...
- 15.6k
2
votes
0
answers
376
views
Reflection formula for the Hurwitz zeta function and odd zeta values
A reflection formula for the Hurwitz zeta function, which does not seem to be well known, uses half of the polynomials generated by $\frac{1}{-1+\sqrt{t-1}\cot(\sqrt{t-1}u)}$. (Look at the sections "...
- 13.4k
2
votes
0
answers
153
views
system of complex number equations
Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
$$(1+2|a_1|^2+2|a_1|^4)a_1+(1+2|a_2|^...
- 309
2
votes
0
answers
187
views
Multiplying three factorials with three binomials in polynomial identity
I have checked the following identity (1) below for $n\leq 40$ with a computer.
Let $(n)_k$ denote the falling factorial $n(n-1)\ldots (n-k+1)$, let
$Z_n=\sum_{k=0}^n (n)_k x^{n-k}$, and finally let
$...
- 621
2
votes
1
answer
751
views
Transfinite sums related to a sequence
Given a sequence $S$ indexed by the finite ordinals, a limit ordinal $\alpha$, and $k \in \mathbb{N}$, define $S_{\alpha+k}$(the extension of $S$ to $\alpha+k$) to be the sum over the products of all $...
- 333
1
vote
0
answers
531
views
Two conjectural identities involving $\zeta(3)$ and the golden ratio $\phi$
Let $\phi$ be the famous golden ratio $\frac{\sqrt5+1}2$, and let
$\zeta(3)=\sum_{n=1}^\infty\frac1{n^3}.$
In 2014, in the paper
Zhi-Wei Sun, New series for some special values of $L$-functions, ...
- 15.6k
1
vote
0
answers
116
views
In search of multiple expressions for a sequence
The sequence $a_n=\sum_{k=0}^n\binom{n}k^24^k$ is listed on OEIS along with a couple of combinatorial interpretations. What interested me at the moment is the plethora of binomial single-sums for the ...
- 43.2k
0
votes
0
answers
244
views
Looking for a combinatorial proof of an identity
I've come up with an interesting combinatorial identity (thanks to P. Belmans who precomputed the numbers and pointed out to me that they correspond to OEIS A002697):
$$
\sum_{i=0}^{n-1}\binom{n+1-i}{...
- 1,810
0
votes
0
answers
302
views
An alternating sum involving a product of binomial coefficients
I encountered the sum below, where $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$ and $d$ are some given positive constants. Does anyone have an idea how to simplify it?
$$
\sum\limits_{k=1}^{d} \frac{(-1)^{k-1}k}...
- 109