Formula to duplicate data in spreadsheet

Question

I have a spreadsheet and would like to use the data it contains for a project I am working on. I would prefer to generate the data mathematically rather than read it in from a .csv file. I haven't been able to figure out the relationship between the data.

Obviously, each number along the diagonal is:

    0.5^n 

and each column has a number of rows in it corresponding to number in the header. It is my belief that the data is derived from the formula in the attached link related to random walks.

$$ p(Y_n= m) = {n \choose (m+n)/2} \frac{1}{2^n} $$

The table below is a small piece of the spreadsheet which extends from 11 to 1000 in the header continuing in the same way.

       10           9           8           7           6         5        4       3      2     1
    1  1.000        1.000       1.000       1.000       1.000     1.000    1.000   1.000  1.000 1.000
    2  0.75390625   0.75390625  0.7265625   0.7265625   0.6875    0.6875   0.625   0.625  0.5   
    3  0.54882813   0.5078125   0.5078125   0.453125    0.453125  0.375    0.375   0.25     
    4  0.34375      0.34375     0.2890625   0.2890625   0.21875   0.21875  0.125            
    5  0.2265625    0.1796875   0.1796875   0.125       0.125     0.0625                
    6  0.109375     0.109375    0.0703125   0.0703125   0.03125                 
    7  0.06542969   0.0390625   0.0390625   0.015625                        
    8  0.02148438   0.02148438  0.0078125                           
    9  0.01171875   0.00390625                              
   10  0.00195313                                   

It is my intention to either calculate these values directly, if possible, or set up a multidimensional array to access the probabilities at each step.

What formula or method can I use to duplicate this data?

Answer 1

After working for an hour or so with the data, for me it seems like the formula is not of the form $$ f(n,m) = { p \choose q} \frac{1}{2^r} $$ where $p$, $q$ and $r$ are some integers that are obtained from some "easy" combination of $n$ and $m$. For example, let's take the data point at $(2,10)$ which is $0.75390625= \frac{193}{256}$. It is worth noting that $193$ is a prime number. The denominator is indeed a power of two. Now we can try to find $p$ and $q$ so that

\begin{equation} { p \choose q} = 193 \times 2^r \Rightarrow {p \choose q} \equiv {0 \mod 193} \end{equation}

Actually, in order for ${p \choose q}$ to be $\equiv {0 \mod p^* }$ for a prime number $p^*$, $p$ has to be larger or equal to $p^*$. Therefore, in order for the equation above be true, $p$ has to be $p \geq 193$. This is much larger than $2$ and $10$.