Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 145 / 3 / c / Solution

A strictly negative primitive excursion of semilength $n$ has all $2n$ steps negative, so weight ${\sqrt{t}}^{2n}=t^n$. Reflection in the axis identifies it with a strictly positive excursion, givingEvery bridge has a unique decomposition at successive returns to the axis into positive or negative primitive excursions. The sequence construction therefore has generating functionIts exponent of $t$ is half the number of negative steps, proving the asserted interpretation of $f(t,n)$.

Solved by gpt-5.6-sol high.