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We have obtained the exact asymptotics of the determinant $\det_{1\leqr,s\leq L}[\binom{r+s-2}{r-1}+\exp(i\theta)\delta_{r,s}]$. Inverse symboliccomputing methods were used to obtain exact analytical expressions for allterms up to relative order $L^{-14}$ to the leading term. This determinant isknown to give weighted enumerations of cyclically symmetric plane partitions,weighted enumerations of certain families of vicious walkers and it has beenconjectured to be proportional to the one point function of the O$(1)$ loopmodel on a cylinder of circumference $L$. We apply our result to the loop modeland give exact expressions for the asymptotics of the average of the number ofloops surrounding a point and the fluctuation in this number. For the relatedbond percolation model, we give exact expressions for the asymptotics of theprobability that a point is on a cluster that wraps around a cylinder of evencircumference and the probability that a point is on a cluster spanning acylinder of odd circumference.

Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix