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I have $n$ coins and each of them shows head with probability $p$ if I throw them separately. Now I throw them one-by-one, and each current throw has a correlation $\rho$ with the previous result. (I hope this is rigorously defined through the definition of correlation coefficient.) After one round of tossing, what is the probability that I got $k$ heads in total?

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Your assumptions imply that either $n=1$ or $p = 1/2$ or $\rho = 1$. Given any of these, it is straightforward to write a recurrence for the probability of getting $k$ heads out of $n$ throws, given that the first coin was $C$ ($C$ is either heads or tails). Using the recurrence, it is then easy to obtain a generating series for the sought-for probabilities. –  Yuval Filmus Feb 26 '11 at 20:28
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Details are all in the answer which used to be here, deleted for lack of interest. –  Yuval Filmus Feb 26 '11 at 20:29

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