Do Bernoulli polynomials know about face vectors? 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?
 A: Following up on i9Fn's suggestion, there does seem to be at least one more case along these lines: $$660F_{11}(n) = 60n^{11} + 330n^{10} + 550n^9 - 660n^7 + 660n^5 - 330n^3 + 50n.$$
This might be related to a genus 70 Riemann surface with 11 embedded buckyballs; see Martin & Singerman From biplanes to the Klein quartic and the Buckyball.  Each buckyball/ buckminsterfullerene has 60 vertices, 90 edges, and 32 faces.  The number 660 arises as both the order of $PSL(2,11)$ and the count of particular triangles.
A: Explicitly, the polynomials are
sage: 12*(bernoulli_polynomial(x,5)-bernoulli(5))                               
12*x^5 - 30*x^4 + 20*x^3 - 2*x
sage: 24*(bernoulli_polynomial(x,7)-bernoulli(7))                               
24*x^7 - 84*x^6 + 84*x^5 - 28*x^3 + 4*x
sage: 60*(bernoulli_polynomial(x,11)-bernoulli(11))                             
60*x^11 - 330*x^10 + 550*x^9 - 660*x^7 + 660*x^5 - 330*x^3 + 50*x

where the scaling factors are the cardinalities of the groups $A_4$, $S_4$, $A_5$.
