Daboul, Siad; Mangaldan, Jan; Spivey, Michael Z.; Taylor, Peter J. The Lah numbers and the \(n\)th derivative of \(e ^{1/x }\). (English) Zbl 1274.05014 Math. Mag. 86, No. 1, 39-47 (2013). 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. Cited in 12 Documents MSC: 05A15 Exact enumeration problems, generating functions 05A19 Combinatorial identities, bijective combinatorics Keywords:Lah numbers; coefficients in the \(n\)th derivative of \(e^{1/x}\); Fàa di Bruno’s formula; Maclaurin series; Poisson distribution; Kummer’s hypergeometric transformation PDFBibTeX XMLCite \textit{S. Daboul} et al., Math. Mag. 86, No. 1, 39--47 (2013; Zbl 1274.05014) Full Text: DOI