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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/03%3A_Generating_Functions/3.03%3A_Exponential_Generating_FunctionsThus ex+e−x=∞∑i=02x2i(2i)!, so that ∞∑i=0x2i(2i)!=ex+e−x2. A similar manipulation shows that \[ \sum...Thus ex+e−x=∞∑i=02x2i(2i)!, so that ∞∑i=0x2i(2i)!=ex+e−x2. A similar manipulation shows that ∞∑i=0x2i+1(2i+1)!=ex−e−x2. Thus, the generating function we seek is ex−e−x2ex+e−x2ex=14(ex−e−x)(ex+e−x)ex=14(e3x−e−x). Notice the similarity to Example 3.2.4.
- https://math.libretexts.org/Courses/Saint_Mary's_College_Notre_Dame_IN/SMC%3A_MATH_339_-_Discrete_Mathematics_(Rohatgi)/Text/6%3A_Additional_Topics/6.1%3A_Generating_FunctionsThere is an extremely powerful tool in discrete mathematics used to manipulate sequences called the generating function. The idea is this: instead of an infinite sequence (for example: 2,3,5,8,12,… )...There is an extremely powerful tool in discrete mathematics used to manipulate sequences called the generating function. The idea is this: instead of an infinite sequence (for example: 2,3,5,8,12,… ) we look at a single function which encodes the sequence. But not a function which gives the n th term as output. Instead, a function whose power series (like from calculus) “displays” the terms of the sequence.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_(Levin)/5%3A_Additional_Topics/5.1%3A_Generating_FunctionsThere is an extremely powerful tool in discrete mathematics used to manipulate sequences called the generating function. The idea is this: instead of an infinite sequence (for example: 2,3,5,8,12,… )...There is an extremely powerful tool in discrete mathematics used to manipulate sequences called the generating function. The idea is this: instead of an infinite sequence (for example: 2,3,5,8,12,… ) we look at a single function which encodes the sequence. But not a function which gives the n th term as output. Instead, a function whose power series (like from calculus) “displays” the terms of the sequence.
