Probability diffusion in the binomial lattice

Given that we can express the diffusion of asset prices in the binomial tree as a summation of the form

The expected stock price at a given time-step can be expressed in terms of the probability, p, of an upward movement, u, of the underlying asset. Conversely the probability of a downward movement in the asset price is denoted as 1-p, where d, is the amount by which the stock moves down. Hence the expected asset price at the first time-step can be summarised as:

Why the vectors? Well I suppose there is a directional, albeit unidirectional, aspect to the diffusion. We are interested in heading towards the future and ascertaining the expected value of the asset at the expiry date.

The obvious question then becomes, “how do we generalize the findings above for a binomial tree with n time-steps?”

The probability diffusion for each time-step appears to follow the summation above. A few tests should be enough to prove whether this is actually the case.