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

1, -14, 86, -3660, -1042202, -247948260, -108448540420, -67825082899288, -56771982322924154, -61577812542004343156, -84012331763021201187180, -140805160243370476949256616, -284390871665315095422337087524

Offset: 0

Author

Richard P. Brent, Dec 08 2018

Keywords

• sign
• frac

Comments

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).

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.

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.

Examples

			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, ...
		

Links

• Richard P. Brent, Table of n, a(n) for n = 0..50
• Richard P. Brent, M. L. Glasser, Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.
• NIST Digital Library of Mathematical Functions, Confluent Hypergeometric Functions

Crossrefs

Cf. A321939, A321940.

Formula

A recurrence is given in Corollary 17 of Brent et al. (2018).