Even binomial sum and constrained monotone maps

Compute \[ \sum_{k=0}^{\lfloor n/2\rfloor}\binom n{2k}. \]

Let \(c_n\) be the number of monotonically increasing functions from \(\{1,2,\ldots,n\}\) to itself satisfying \(f(i)\le i\) for every \(i\). Find a recurrence relation satisfied by \((c_n)_{n\ge1}\).