# Coincidence between coefficients of tanh(tan(x/2)) and Chow ring computations?

In "Kodaira–Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes" by Bershadsky, Cecotti, Ooguri, and Vafa (arxiv) on pg. 96 appear the two numbers $$5760$$ and $$1451520$$ related to integrals over moduli spaces of Riemann surfaces of genus $$g=2$$ and genus $$3$$ computed from certain quantities related to Chow rings. The associated numbers $$2880$$ and $$90720$$ in Vafa et al. are in Table 10, pg. 418, of "Chow rings of moduli spaces of curves I: The Chow ring of $$\overline{\mathcal{M}}_3$$" by C. Faber (JSTOR) and give $$2 \cdot 2880 = 5760$$ and $$2^4 \cdot 90720 = 1451520.$$

The numbers appear in OEIS A013524 as the denominators of the expansion

$$\tanh(\tan(x/2)) = \frac{1}{2}x - \frac{1}{480} x^5 - \frac{1}{5760} x^7 - \frac{1}{1451520} x^9 + \frac{13}{9676800} x^{11} + \cdots.$$

Not being familiar with the concepts, I was not able to come up with any references that presented results for any other genera.

Is this mere coincidence; that is, have computations for higher order genera been done, and, if so, are they included in this expansion?

The integrals $$\int_{\overline{\mathcal{M}}_g} \lambda_{g-1}^3=\frac{|B_{2g}|}{2g}\frac{|B_{2g-2}|}{2g-2}\frac{1}{(2g-2)!}$$ were computed by Faber and Pandharipande in Theorem 4 in their paper Hodge integrals and Gromov-Witten theory here. They also provide references to derivations of the formula from string theory. I think one should be able to make the connection to the expansion in the question via manipulations with the Bernoulli numbers.

• Nice extension of results. For $g = 5$, there is no integer factor that gives an equality between twice the reciprocal of the Bernoulli expression and the denominator of the $x^{13}$ coefficient in the expansion, so the expansion does not serve as a generating function. Commented Jul 8 at 9:33
• This formula and its context appear in section 6. Dualities of "Geometry and physics" by Michael Atiyah, Robbert Dijkgraaf and Nigel Hitchin (royalsocietypublishing.org/doi/full/10.1098/…). Commented Jul 9 at 22:11

I extracted the denominators of the coefficients of the series expansion from the seventh power

l1=Denominator[SeriesCoefficient[Tanh[Tan[x/2]],{x,0,#}]&/@Range[7,2n+3,2]]

and compared them with the denominators of the coefficients of the expression in @SamirCanning's answer

l2=Denominator[Table[Abs[BernoulliB[2g]BernoulliB[2g-2]]/((2g-2)!*2g(2g-2)),{g,2,n}]].

When n=40, the ratio l2/l1 yields

$$\small\left\{2,2,\frac29,39,\frac16,\frac{68}9,\frac25,\frac1{312},\frac4{135},\frac{28}{13},\frac{20}{63},290,\frac{504}5,\frac8{273},\frac2{27},\frac{8547}{104},\frac43,\frac{41}{84672},\frac{91}{1320},\frac1{4165},\frac{88}{3675},1176,\frac{20}{30537},\frac{371}{2178},\frac{15912}{301},\frac{88}3,\frac{88}{1215},4636,\frac2{715},\frac{16}{825},\frac{2002}{17},\frac{143}{68880},\frac{2516}9,\frac{2609166}{11},\frac{10}{272675403},\frac{55}{702},760760,1287,\frac1{615230}\right\}$$ and it must be remarked that most of the prime factors in l1 cancel out with those in l2, leading to many ratios being surprisingly simple relative to the number of digits of either denominator. However I cannot mathematically prove this yet.

• Just out of curiosity, A) one could look for some regularities such as the denominators of those fractions whose numerators are 2 in the OEIS (quite a lot of hits for those you display--search 9; 5; 27; 715); B) use the Bernoulli polynomials instead of the Bernoulli numbers for a fit, similar to the analysis in "Moduli spaces of flat connections on 2-manifolds, cobordism, and Witten's volume formulas " by Meinerenken and Woodward; Commented Jul 8 at 17:51
• C) transform and explore relations to the sawtooth fct. $\mathrm {sawtooth} (\theta )=2\arctan(\tan({\frac {\theta }{2}}))$, essentially the periodic Bernoulli fct. of first order, whose number theoretic properties are discussed in "A Friendly Invitation to Fourier Analysis on Polytopes" by Robbins and "Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra" by Beck and Robbins. Commented Jul 8 at 17:54