$e^{e^x}$在0的5次导

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$$\left.\frac{d^5}{dx^5}e^{e^x}\right|_{x=0}=52e$$ 因为5个元素的划分个数为52。Partition of a set image.svg (113.97 KB, Downloads: 97)

2023-7-24 04:26 Upload

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You prefer calculus? Fine, compute the 5th derivative of f(g(x)), or evaluate the 5th derivative of e^e^x at 0 (52e). Then figure out why that was related to the set partitions.

Notes on partitions and their generating functions
Exponential generating function of partitions of set [n]

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划分	个数
1个集合	1
2个集合分别有1、4个元素	C(5,1) = 5
2个集合分别有2、3个元素	C(5,2) = 10
3个集合分别有3,3,1或2,2,1个元素	25
4个集合分别有1,1,1,2个元素	C(5,2)=10
5个集合分别有1,1,1,1,1个元素	1

总计52

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