A Conjectured Integer Sequence Arising From the Exponential Integral

Let f(0)(z) = exp(z/(1 - z)), f(1)(z) = exp(1/(1 - z))E-1(1/(1 - z)), where E-1(x) = integral(infinity)(x) e(-t)t(-1) dt. Let a(n) = [z(n)] f(0)(z) and b(n) = [z(n)]f(1)(z) be the corresponding Maclaurin series coefficients. We show that a(n) and b(n) may be expressed in terms of confluent hypergeometric functions. We consider the asymptotic behaviour of the sequences (a(n)) and (b(n)) as n -> infinity, showing that they are closely related, and proving a conjecture of Bruno Salvy regarding (b(n)). Let rho(n) = a(n)b(n), so Sigma rho(n)z(n) = (f(0 )circle dot f(1))(z) is a Hadamard product. We obtain an asymptotic expansion 2n(3/2) rho(n) similar to - Sigma d(k)n(-k) as n -> infinity, where d(k) is an element of Q, d(0) = 1. We conjecture that 2(6k)d(k) is an element of Z. This has been verified for k <= 1000.