# Tagged Questions

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### Maximize combinatorial sum for boolean function

I am trying to maximize the function $$S(f)=\sum_{j=0}^{n-\frac{n-1}{t}}(-1)^j{n-\frac{n-1}{t}\choose{j}}\sum_{i=0}^{\frac{n-1}{t}}(-1)^{f(i-j)}(t-1)^i{\frac{n-1}{t}\choose{i}}$$ for a function ...
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### Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$ \begin{array}{cccccccccc} 1 & = & ...
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### (Asymptotics of) Sum involving alternating sign Chu-Vandermonde

While considering eigenvalues of a certain Cayley graph, I came across the following sum: $$\sum_{r=0}^{d}\sum_{i=0}^{r} (-1)^{i} \binom{w}{i}\binom{n-w}{r-i}$$ where $d$, $w$, and $n$, are all ...
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### alternating sum of binomial coefficients

I would like to know a closed formula for $\sum_{j=0}^{p-n } (-1)^j\binom{n^2}{p-n-j} \binom{n+j-1}j\binom{2n+j}{n+j+1}$, especially in the case $p$ is near $n^2/2$. Similarly, I would like a closed ...
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### Polynomials with presumably positive coefficients

The $q$-Pochhammer symbol $(q) _ 0:=1$ and $(q) _ n:=\prod _ {j=1}^n (1-q^j)$ for $n > 0$ is clearly a polynomial in $q$ which has both positive and negative coefficients when $n>0$. The ...
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### Is there a closed formula for the generating function of some trinomial coefficients?

We learn in calculus how to obtain a sum of binomial coefficients \frac{(2d)!}{(d!)^2} in terms of a generating function $\sum_{d \geq 0} \frac{(2d)!}{(d!)^2} x^d$ by the Taylor series of ...
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### Sums of binomials with even coefficients

While looking for a closed form of a expression I worked myself to a formula that resembles the Vandermonde convolution, but is summed over even binomial coefficients only. ...
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### non negative integer solutions : Diophantine Equations [closed]

I want to know the exact number of non-negative integer solutions of a1x1+a2x2+...akxk = n ... I know that it is the co-efficient of x^n in (1-x^a1)^-1 * (1-x^a2)^-1 * ... (1-x^ak)^-1 ... but whats ...
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### Binomial coefficient in Andrews' partition book

First of all, i think MathOverflow is a very great community to discuss math, either basic or advanced, and i'm glad to participate here. It's my first post, so i'm sorry if i did anything wrong, and ...