Question about A051002 in OEIS

Question

I have a question about sequence https://oeis.org/A051002 in the OEIS database with first elements

1, 1, 244, 1, 3126, 244.

I searched however for

1,-1, 244, -1, 3126, -244

a sequence described by a Generating Function in Exercise 1.15 of Apostol's MFDSNT.

My question is about the following reference in the sequence description:

"T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25, Exercise 15 (corrected)."

What is meant with (corrected) in this reference? Is there an error in the exercise, perhaps? I searched for an Errata but there is none, so I don't think so. Besides both GF's in the exercise result in

1,-1, 244, -1, 3126, -244.

Why the reference to Apostol Exercise 1.15 at all, I wonder.

Answer 1

The OEIS sequence A051002 is "Sum of 5th powers of odd divisors of n." The reference to Apostol with "Exercise 15 (corrected)" is not clear here, but certainly refers to the fact that the exercise refers to

and let $$ F(x) = \sum_{n=1\\ n\,\text{odd}}^\infty \frac{n^5 x^n}{1+x^n}.$$

whereas the OEIS entry has a formula

Compare this with the Apostol reference which gives the g.f. of a(n)*(-1)^(n+1). - Wolfdieter Lang, Jan 31 2017

which explains that there is an alternating sign difference in the two sequences. Thanks to your question, I have modified the OEIS entry to remove the "(corrected)".