> #ATTENDANCE QUIZ 23 ; > #THE NUMBER OF ATTENDANCE QUESTIONS WERE: 3 ; > ; > #Q1 ; > #Question 1: Use Lagrange Inversion Formula to find an explicit expression for the coeff of x^n in the power series of u(x) that satisfies the functional equation: > ; > #u(x) = x*(1+u(x))^2 ; > ; > ; > #u(x) = x*(1+u(x))^3 ; > ; > ; > ; > # for any positive integer k u(x) = x*(1+u(x))^k ; > ; > ; > #Q2 ; > #Consider the function above. (i) Draw this directed graph (ii) Find the doubly-rooted tree that is outputted by the joyal bijection ; > ; > ; > ; > ; > #Q3 ; > #Finish this. Find the Pruffer Code (of length 11) of the original tree. ;