  We derive a Laurent series expansion for the structure coefficients appearing
in the dual basis corresponding to the Kauffman diagram basis of the
Temperley-Lieb algebra $\text{TL}_k(d)$, converging for all complex loop
parameters $d$ with $|d| > 2\cos\big(\frac{\pi}{k+1}\big)$. In particular, this
yields a new formula for the structure coefficients of the Jones-Wenzl
projection in $\text{TL}_k(d)$. The coefficients appearing in each Laurent
expansion are shown to have a natural combinatorial interpretation in terms of
a certain graph structure we place on non-crossing pairings, and these
coefficients turn out to have the remarkable property that they either always
positive integers or always negative integers. As an application, we answer
affirmatively a question of Vaughan Jones, asking whether every Temperley-Lieb
diagram appears with non-zero coefficient in the expansion of each dual basis
element in $\text{TL}_k(d)$ (when $d \in \mathbb R \backslash
[-2\cos\big(\frac{\pi}{k+1}\big),2\cos\big(\frac{\pi}{k+1}\big)]$).
Specializing to Jones-Wenzl projections, this result gives a new proof of a
result of Ocneanu, stating that every Temperley-Lieb diagram appears with
non-zero coefficient in a Jones-Wenzl projection. Our methods establish a
connection with the Weingarten calculus on free quantum groups, and yield as a
byproduct improved asymptotics for the free orthogonal Weingarten function.
