Pretending for the sake of argument that I don't see any regularities in your sequence that I wouldn't expect from genuinely random coin flips (it actually looks to me more human-generated, but with only n=36 I'm not very confident of that): the odds are pretty much the same as the prior odds that you'd actually flip a coin 36 times rather than just writing down random-looking Hs and Ts.

You said something interesting there, and then skipped right past it. That's the substance of the question. You don't get to ignore those regularities; they do, in fact, affect the probabilities. Saying that they don't appear in the ratio of Bayes' Rule is... well, misusing Bayes' Rule to discard meaningful evidence.

The only formula I wrote down was "2^-n times Pr(Alice cheated)" and those probabilities are definitely not greater than 1. Would you care to be more explicit?

2^(-n) approaches 1 as n approaches infinity, but for any finite n, is greater than 1. Multiply that by a probability of 1, and you get a probability greater than 1. [ETA: Gyah. It's been too long since I've done exponents (literally, ten years since I've done anything interesting). You're right, I'm confusing negative exponents with division in exponents.]