Find the generating function to determine the number of ways to pick k objects from n objects when repetitions are allowed and the ith object appears i or 2i times for 1 ≤ i ≤n.

Let n be a positive integer. Give a recursive algorithm for computing the sum of the first n positive integers by filling in the boxes in the procedure below with the expressions n, n – 1 or 1. Note: You have to use some expressions more than once, but each box must contain only one of those expressions. procedure sum : positive integer) if then sum 1 else sum + sum

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Let S be the subset of the set of ordered pairs of integers defined recursively by Basis step: (0,0) = S F Recursive step: If (a, b) = S, then (a, b + 1) = S, (a + 1, b + 1) = S, and (a + 2, b + 1) = S. List the elements of S produced by the first four applications of the recursive definition. Enter your answers in the form (a₁, b₁), (a2, b2),..., (an, bn), in order of increasing a, without any spaces. The first application of the recursive step adds (Click to select) ✓to S. The second application of the recursive step adds (Click to select) The third application of the recursive step adds (Click to select) The fourth application of the recursive step adds (Click to select) to S. ✓to S. ✓to S.