Alice, Bob, Carol, and Dave are sharing n cookies. Alice wants any positive number of cookies. Bob wants the same number of cookies as Alice. Carol wants any number of cookies. Dave wants at most 1 cookie (≤ 1). Together they must finish all n. Find a closed form of the generating function f(x) that describes the number of ways of sharing cookies. Rose and Colin are sharing n cookies. Rose wants any number of cookies. Colin wants at least 2 cookies (≥ 2). Together they must finish all n. Show that for any n there is the same number of ways for Rose and Colin to share the cookies as Alice, Bob, Carol, and Dave.

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