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 Mar 28 comment Recursively defined numbers vs geometric series @JasonStarr. I do mean that $h$ is a rational function of $d_1$ and $d_2$. It can be assumed to be symmetric. In the context of Kontsevich's formula $h(d_1,d_2)\!=\!f(d_1\!+\!d_2,d_1d_2)/g(d_1\!+\!d_2)$ for some polynomials $f$ and $g$. Since the number of $(d_1,d_2)$ with $d_1\!+\!d_2\!<\!2d(\epsilon)$ is finite, there is a bound on the corresponding $h(d_1,d_2)$, but it depends on $\epsilon$. Mar 28 comment Recursively defined numbers vs geometric series @JasonStarr. . With a specific and somewhat ugly $h$, this recursion is equivalent to Kontsevich's formula for $2^dn_d/(3dâˆ’1)!$; there is more in my post from 2 days ago. While only this specific $h$ is relevant in this context, I find the general question interesting and more natural. I realize that $n_1$ can equivalently be taken to be any positive real. Mar 28 comment Recursively defined numbers vs geometric series @JasonStarr. Since this sequence is bounded above and below, I do not find it too surprising that it has many increasing subsequences. Of course, it could be just decreasing. This would be fine for my purposes too, since I actually want to know that $\sqrt[d]{n_d}$ has a limit (in relation to my question from 2 days ago). Mar 28 asked Recursively defined numbers vs geometric series Mar 28 comment Asymptotic analysis of generating functions @JayPantone. The paper is hep-th/9412175. My $F$ is (2.54) after normalizing by $a^d$, with $a$ as in (2.59). The expansion is (2.56), along with (2.58). The limsup statement is a weaker version of (2.59). The ODE is (2.55). I'll post a note on arXiv on this with related observations and speculations from 3 years ago soon, now that I know that even the limsup conclusion is not a triviality. One of the speculations is my next MO posting. Mar 28 comment Asymptotic analysis of generating functions @GeraldEdgar. Thank you for the suggestion; it looks like a nice book. Unfortunately, none of the three cases of Chapter 5 covers polylogarithmic singularities. Thm 5.3.1 is about algebraic singularities. On the other hand, looking at this book suggested to me how to think about the first part of my question and to realize that the answer to it is no. There could be a subsequence of $a_d$ with values of say $d^{-3}$. This would mean the limsup is infinite, but the expansion could be valid (if these value appear rarely enough). So, even the weaker conclusion requires some clever use of the ODE. Mar 25 asked Asymptotic analysis of generating functions Nov 19 awarded Commentator Nov 19 comment Reference request for cohomology of coverings @BenWieland. It sounds like there is no reasonable statement along the lines of the original question then. This is very enlightening. Many thanks, Ben, as well as Matthias and Dan. Nov 18 comment Reference request for cohomology of coverings This answer was meant to give an alternative description of Matthias's answer for cyclic coverings. As Ben points out, this does not suffice for the inductive step and there is no hope of extending this construction even to ${\mathbb{Z}}^2$-coverings without additional assumptions. Nov 18 comment Reference request for cohomology of coverings Is there a statement in the spirit of the original one which is true? It is true if the rank of $G$ is at most one. Is it true if the abelianization of the commutator subgroup of $\pi_1(B)$ is finite? or perhaps even the commutator subgroup itself is finite? Nov 18 comment Reference request for cohomology of coverings Thank you, Ben. I think I see where my thinking with the induction went wrong. $H^*(\hat{B})^{\mathbb Z}$ comes from $H^*(\hat{B}/{\mathbb Z})$, but only $H^*(\hat{B}/{\mathbb Z})^{\mathbb Z}$ comes from $H^*(\hat{B}/{\mathbb Z}^2)$. More relevantly, one needs to show that $H^*(\hat{B})^{\mathbb{Z}^2}$ comes from $H^*(\hat{B}/{\mathbb{Z}})^{\mathbb{Z}}$, and there was nothing in the inductive step about some auxiliary group. Nov 16 answered Reference request for cohomology of coverings May 31 awarded Critic Apr 14 comment On the proof of Robert Lipshitz's formula on Maslov index. Why not just e-mail, Robert? Mar 31 answered Difference between parallel transport and derivative of the exponential map Feb 19 awarded Enthusiast Jan 29 answered Analytic curve on Riemann surface Jan 28 revised Tubular neighborhoods of chains added 36 characters in body Jan 28 revised Tubular neighborhoods of chains added 156 characters in body; deleted 103 characters in body