cp's OEIS Frontend

If r(n) = A067764(n)/A067653(n) then r(n)/(exp(2*sqrt(n))/(2*n^(3/4)*sqrt(Pi*e))) has an asymptotic expansion in ascending powers of 1/sqrt(n) whose coefficients are rational numbers 1, -5/48, etc. The sequence gives the denominators of these rational numbers.

Another expression for r(n), n > 0, is r(n) = M(n+1,2,1)/e, where M(a,b,z) = 1F1(a;b;z) is a confluent hypergeometric function (Kummer function).

The same rational numbers, except for signs, occur in the asymptotic expansion of the Maclaurin coefficients of exp(1/(1-x))*E1(1/(1-x)), where E1(x) is an exponential integral. See Lemmas 1-2 and Theorem 5 of the preprint by Brent et al. (2018).

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.