Oops, I should have said "numerators of $q$-Bernoulli numbers". These rational functions were introduced by Carlitz in Carlitz, L. q-Bernoulli numbers and polynomials. Duke Math. J. 15, (1948). 987–1000. They are available in sage as follows: "from sage.combinat.q_bernoulli import q_bernoulli" then "q_bernoulli(20).numerator()"

I have now added an image where one can compare the decay with the decay of a similar Gaussian.

It seems to me that the function $sin(x)/x$ is not decreasing fast enough. Its graph does not look similar, and has long tails and many visible oscillations.

The polynomials are numerators of $q$-Bernoulli polynomials (and variants of that).

The picture is the plot of the list of coefficients of one polynomial (in a familly of polynomials indexed by the integers). This is essentially a sequence of points, one with coordinates $(i,c_i)$ for each monomial $c_i x^i$.

There are a few words at the end of section 4 of Borcherds' article "Modular Moonshine II" (math.berkeley.edu/~reb/papers/modular2/modular2.pdf) about $E_8(3)$ and the Thompson group. But this concerns modular representations of Th.

Hum, maybe there is also a 240 in the coefficient 708938760, as it is the coefficient of $q^{16}$ ?

The next coefficient 708938760 seems to be harder to decompose..

One also has 91951146 = 779247 + 91171899.