Journal of Integer Sequences, Vol. 22 (2019), Article 19.4.7

A Conjectured Integer Sequence Arising From the Exponential Integral

Richard P. Brent
Australian National University
ACT 2601

M. L. Glasser
Clarkson University
Potsdam, NY 13699-5820

Anthony J. Guttmann
School of Mathematics and Statistics
The University of Melbourne
Vic. 3010


Let $f_0(z) = \exp(z/(1-z))$, $f_1(z) = \exp(1/(1-z))E_1(1/(1-z))$, where $E_1(x) = \int_x^\infty e^{-t}t^{-1}{\,d}t$. Let an = [zn]f0(z) and bn = [zn]f1(z) be the corresponding Maclaurin series coefficients. We show that an and bn may be expressed in terms of confluent hypergeometric functions.

We consider the asymptotic behaviour of the sequences (an) and (bn) as $n \to \infty$, showing that they are closely related, and proving a conjecture of Bruno Salvy regarding (bn).

Let $\rho_n = a_n b_n$, so $\sum \rho_n z^n = (f_0{\,\odot}f_1)(z)$ is a Hadamard product. We obtain an asymptotic expansion $2n^{3/2}\rho_n \sim -\sum d_k n^{-k}$ as $n \to \infty$, where $d_k\in{\mathbb Q}$, d0=1. We conjecture that $2^{6k}d_k \in {\mathbb Z}$. This has been verified for $k \le 1000$.

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(Concerned with sequences A000012 A000262 A067653 A067764 A073003 A201203 A293125 A321937 A321938 A321939 A321940 A321941 A321942.)

Received December 14 2018; revised versions received December 15 2018; May 27 2019. Published in Journal of Integer Sequences, August 15 2019.

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