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A321941 Scaled numerators in the asymptotic expansion of the Maclaurin coefficients in a Hadamard product involving the exponential integral.

%I A321941 #17 Dec 09 2018 06:42:44
%S A321941 1,-14,86,-3660,-1042202,-247948260,-108448540420,-67825082899288,
%T A321941 -56771982322924154,-61577812542004343156,-84012331763021201187180,
%U A321941 -140805160243370476949256616,-284390871665315095422337087524
%N A321941 Scaled numerators in the asymptotic expansion of the Maclaurin coefficients in a Hadamard product involving the exponential integral.
%C A321941 a(n) is the numerator of the rational number called r_n in Brent et al. (2018). It is conjectured that r_n is an integer, so the denominators should all be 1 (this has been verified for n <= 1000). A stronger conjecture is given in Remark 12 of Brent et al. (2018).  It is known that n!*r_n is an integer, see Theorem 18 of Brent et al. (2018).
%C A321941 d_n = r_n/64^n can be written as a signed convolution of the rational numbers c_n = A321939(n)/A321940(n), see Corollary 10 of Brent et al. (2018). For example, c_0 = 1, c_1 = -5/48, c_2 = -479/4608, and d_1 = c_0*c_2 - c_1*c_1 + c_2*c_0 = -7/32.
%C A321941 d_k = r_k/64^k is the k-th coefficient in the asymptotic expansion of (2/e)*n^(3/2)*Gamma(n)*M(n+1,2,1)*U(n,0,1), where M and U denote confluent hypergeometric functions (Kummer functions), see Brent et al. (2018), Sections 3 and 5.
%H A321941 Richard P. Brent, <a href="/A321941/b321941.txt">Table of n, a(n) for n = 0..50</a>
%H A321941 Richard P. Brent, M. L. Glasser, Anthony J. Guttmann, <a href="https://arxiv.org/abs/1812.00316">A Conjectured Integer Sequence Arising From the Exponential Integral</a>, arXiv:1812.00316 [math.NT], 2018.
%H A321941 NIST Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/13.2">Confluent Hypergeometric Functions</a>
%F A321941 A recurrence is given in Corollary 17 of Brent et al. (2018).
%e A321941 The asymptotic expansion (defined in Corollary 10 of Brent et al. (2018)) has coefficients 1, -7/32, 43/2048, -915/65536, ...  Multiplying by consecutive powers of 64 gives 1, -14, 86, -3660, ...
%Y A321941 Cf. A321939, A321940.
%K A321941 sign,frac
%O A321941 0,2
%A A321941 _Richard P. Brent_, Dec 08 2018