Have some problems that I have no idea how to solve problem (a).

Homework5.pdf 1 / 1 (f: 1 E; {a} Let an be the number of compositions of n into odd parts where each part is colored either red or blue ( for example a1 = 2 since you can write 1 as either a red 1 or a blue 1). Give the ordinary generating function 22:0 aux\". ( b } Let b" be the di'erence between the number of compositions of in into an even number of parts of size 1 and 2, and the number of compositions of n into an odd number of parts of size 1 and 2. Give a closed form for the ordinary generating function 220:0 bnx". Use this to give an explicit formula for bn. (e) Let on = nC'n where OH is the Catalan number. Give an explicit formula for the ordinary generating function 22:0 cnx". [Hint' you might want to consider di'erentiating the generating function for the Catalan numbers. ] ((1) Let an be the number of set partitions of [n] into sets of size exactly 2 [Note: (in will be 0 ifn is odd]? Find a formula for tin, and use it to obtain an explicit form for the exponential generating function 220:0 damn/\"L .. (e) Let en be a sequence with en+2 = n+1 + men for all n 2 0. Give a dierential equation satised by the exponential generating function E(x) = 22:0 enxnl. + Question 3 (Extra credit, 1 point). Approximately how much time did you spend on this