This question is grounded firmly in numerology. It originates in an observation about some Bernoulli polynomials and the regular icosahedron. Let $F_{k+1}(n)=\sum_{i=1}^n i^k$ be the sum of the first n kth powers of positive integers. We know that $F_k$ is a polynomial of degree $k$ that is related to Bernoulli polynomials via an affine-linear substitution for $n$. The observation here concerns $F_5$ and $F_7$. Specifically, we see $$60F_5(n)=12n^5+30n^4+20n^3-2n$$ and $$168F_7(n)=24n^7+84n^6+84n^5-28n^3+4n.$$ Notice that the coefficients of $60F_5$ coincide with the face vector of a regular icosahedron (20 triangles, 30 edges, 12 vertices). Also, the coefficients of $168F_7$ nearly coincide with the face vector of a surface related to the Klein quartic. This is a "regular" triangulation of a surface of genus 3 that has 24 vertices, 84 edges, and 56 triangles. Notice that the sum of the coefficients of $168F_7$ on the terms of degrees 5 and 3 is 56. The numbers 60 and 168 are significant because they are the orders of a couple of groups that act on these polyhedra.

Is this a coincidence? Are there similar phenomena relating higher-degree Bernoulli polynomials to regular triangulations of other surfaces? Should I lay off the moonshine?

This question is grounded firmly in numerology. It originates in an observation about some Bernoulli polynomials and the regular icosahedron. Let $F_{k+1}(n)=\sum_{i=1}^n i^k$ be the sum of the first $n$ $k$th powers of positive integers. We know that $F_k$ is a polynomial of degree $k$ that is related to Bernoulli polynomials via an affine-linear substitution for $n$. The observation here concerns $F_5$ and $F_7$. Specifically, we see $$60F_5(n)=12n^5+30n^4+20n^3-2n$$ and $$168F_7(n)=24n^7+84n^6+84n^5-28n^3+4n.$$ Notice that the coefficients of $60F_5$ coincide with the face vector of a regular icosahedron (20 triangles, 30 edges, 12 vertices). Also, the coefficients of $168F_7$ nearly coincide with the face vector of a surface related to the Klein quartic. This is a "regular" triangulation of a surface of genus 3 that has 24 vertices, 84 edges, and 56 triangles. Notice that the sum of the coefficients of $168F_7$ on the terms of degrees 5 and 3 is 56. The numbers 60 and 168 are significant because they are the orders of a couple of groups that act on these polyhedra.

Is this a coincidence? Are there similar phenomena relating higher-degree Bernoulli polynomials to regular triangulations of other surfaces? Should I lay off the moonshine?