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46 votes
0 answers
3k views

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 ...
Peter Shor's user avatar
  • 6,342
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 \...
Amritanshu Prasad's user avatar
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^...
Alexander Burstein's user avatar
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 ...
Zhi-Wei Sun's user avatar
  • 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 ...
Zhi-Wei Sun's user avatar
  • 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 ...
Zhi-Wei Sun's user avatar
  • 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}(\...
Abhishek Halder's user avatar
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 ...
Mike Spivey's user avatar
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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 ...
T. Amdeberhan's user avatar
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+\...
T. Amdeberhan's user avatar
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{...
Zhi-Wei Sun's user avatar
  • 15.6k
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 ...
T. Amdeberhan's user avatar
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 $1/\pi$ ...
Zhi-Wei Sun's user avatar
  • 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 ...
T. Amdeberhan's user avatar
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}{...
Anton Fonarev's user avatar
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}...
sdd's user avatar
  • 109