Hello, fellow colleagues in the Julia community.

This question may not be a good fit for the Julia Discourse, but any help is appreciated.

I was exploring Analytic Combinatorics (by Flajolet & Sedgewick). The part about binary words (words over a binary alphabet). contains the following generating function. in eq. 54 (pg. 52)

The following part in page 53 (equation without number) describes the probabilities associated with the maximum length of runs of consecutive letters:

I used SymPy.jl to implement it:

```
using SymPy
#using FastRationals
@vars z r n
# Binary words that never have more than r consecutive identical letters is found to be (set α = β = r)
# Flajolet pag. 52
w_rr(r,z) = (1-z^(r+1))/(1-2z+z^(r+1)) # OGF
w_rr(r,z) = sum(z^x for x in 0:r)/(1 - sum(z^x for x in 1:r)) # Alternate form
function n_words(r)
coefs = collect(series(w_rr(r,z),z,0,r+1),z)
coefs.coeff(z,r)
end
function p_k(k,n)
#a = FastRational{Int128}(1/(2^n))
#a = Rational{BigInt}(1/(2^n))
a = 1/(2^n)
result_sym = a*(n_words(k) - n_words(k-1))
return(result_sym)
end
```

Testing for `k=6`

and verifying the Taylor series for `r=15`

:

```
julia> p_k(6,n)
-n
32⋅2
julia>series(w_rr(15,z),z,0,15+1)
2 3 4 5 6 7 8 9 10 11 12 13 14
1 + 2⋅z + 4⋅z + 8⋅z + 16⋅z + 32⋅z + 64⋅z + 128⋅z + 256⋅z + 512⋅z + 1024⋅z + 2048⋅z + 4096⋅z + 8192⋅z + 16384⋅z + 32768
15 ⎛ 16⎞
⋅z + O⎝z ⎠
```

My problems:

1 - Running `p_k(6,200)`

directly returns ∞ or another undefined value, even in with `FastRational{Int128}(1/(2^n))`

or `Rational{BigInt}(1/(2^n))`

that are commented out. I can run `p_k(6,n)`

passing symbolic `n`

and then evaluate the output directly in the REPL (`julia>32*2^(-200)`

) though.

2 - The result seems to be in accordance with the formula `(1/2^n)*([z^6]W<6,6> - [z^5]W<5,5>)`

, but it is quite different from ‘template’ for k=6. The coefs. in the Taylor expansion seem to be just `[z^n]W<k,k> = 2^n`

. Something must be wrong, since monotonically increasing values for k would not give the `P(K)`

distribution presented in the table for integers k in [1,12].

3 - I intend to calculate probabilities and test hypotheses about structures in a recurrence matrix (trapping time, vertical, diagonal), which are also binary words. Any thoughts on that?