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  "body": "\n## Definition\n\nLet $K$ be a field of characteristic $0$, for example $\\mathbb{Q}$ or $\\mathbb{R}$, and let $(a_n)_{n \\ge 0}$ be a sequence in $K$.\nThe exponential generating function of $(a_n)$ is the [[formal power series|072W67KFYK]]\n$$\\hat{A}(x) = \\sum_{n \\ge 0} a_n \\frac{x^n}{n!} \\in K[\\![x]\\!].$$\n\n## Remarks\n\nThe constant sequence $1$ has the exponential generating function $e^x = \\sum_n x^n / n!$.\nFor exponential generating functions, the product corresponds to the binomial convolution $c_n = \\sum_{k=0}^{n} \\binom{n}{k} a_k b_{n-k}$.\nThey suit labelled structures. For example, the [[derangement numbers|03WY4HHMT1]] have the exponential generating function $\\frac{e^{-x}}{1 - x}$.\n\n## Explanations\n\n### L1 human\n\nAn exponential generating function divides the $n$-th term by $n!$ before it puts it in the series.\nThis makes the series of fast-growing sequences, such as $n!$, easy to handle: the sequence $n!$ gives $\\frac{1}{1 - x}$.\n",
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