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The Lah numbers and the \(n\)th derivative of \(e ^{1/x }\). (English) Zbl 1274.05014

Summary: We give five proofs that the coefficients in the \(n\)th derivative of \(e^{1/x}\) are the Lah numbers, a triangle of integers whose best-known applications are in combinatorics and finite difference calculus. Our proofs use tools from several areas of mathematics, including binomial coefficients, Fàa di Bruno’s formula, set partitions, Maclaurin series, factorial powers, the Poisson probability distribution, and hypergeometric functions.

MSC:

05A15 Exact enumeration problems, generating functions
05A19 Combinatorial identities, bijective combinatorics
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