# Tagged Questions

108 views

### A self-convolution formula that counts bracket expressions

Problem: Consider an alphabet of size $m+2$, consisting of the two bracket symbols $\ [ \ ] \$ plus $m$ non-bracket symbols ($m \ge 0$). Define $f_m(n)$ to be the number of length-$n$ strings on this ...
117 views

### Probability with bullets and walls

There are two shooters with different guns and bullets. Each shooter shoots a bullet to a different target hanging on a wall. The hit of each bullet follows a normal distribution centered on its ...
58 views

### Solving $F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i})$?

I need to find $F_{n}$ in : $$F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) , F_0 = 0 , n>=2$$ This equation screams convolution , I think , but I find it as a quite long solution sometimes. ...
Lets define discrete $f_N(i) = 1,\space i = 1...N$ I need to find $G_N^m = \underbrace {f_N * f_N * ... * f_N}_{m}$ For example $G_6^3$ have value (1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1) , ...
This operation is similar to discrete convolution and cross-correlation, but has binomial coefficients: $$f(n)\star g(n)=\sum_{k=0}^n \binom{n}{k}f(n-k)g(k)$$ Particularly, a^n\star ...