Consecutive Occurrences in Roulette

Hi I was working the math and was wondering if anyone could verify the equation I've come up with for the probability of C consecutive losses in s spins of roulette.
Let:
p = probability of winning
q = probability of losing
C = consecutive losses experienced
s = number of spins

(s-C+1)
( k ) * ( sum (from k=0 to s) [q^(s-k) * p^k] - sum (from k=0 to C+1) [q^k * p^(s-k)] )

where the fist part are the binomial coefficients or equivalently the "s-C+1"th row of Pascal's Triangle and k is the 1st, 2nd, 3rd, etc. number in that row.

I see no error in your logic.

P(j,k)=P(j-1,k)+Q(i,k)*DELTA
This is from Pacomartins thread
https://wizardofvegas.com/forum/questions-and-answers/math/4855-ask-the-wizard-correction/2/

Also a Markov Chain approach, the topic of the thread.

This streak calculator has even more links to different methods
http://www.pulcinientertainment.com/info/Streak-Calculator-enter.html

added:
The above methods actually calculate the cdf (cumulative)
There is one method that does the pdf

Occurrence and Nonoccurrence of Random Sequences:
Comment on Hahn and Warren (2009)
has the proofs and examples.
I do not know if that paper is a free pdf.
M daughter has it.