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The Lah Numbers and the n th Derivative of e 1/ x

2013; Taylor & Francis; Volume: 86; Issue: 1 Linguagem: Inglês

10.4169/math.mag.86.1.039

1930-0980

Siad Daboul, Jan Mangaldan, Michael Z. Spivey, Peter J. Taylor,

Analytic Number Theory Research

SummaryWe give five proofs that the coefficients in the nth derivative of exp(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, Faà di Bruno's formula, set partitions, Maclaurin series, factorial powers, the Poisson probability distribution, and hypergeometric functions.