Identify power series with coeffs. that are palindromic polynomials of a param.

[jsm10]

Can someone help me to identify the type of power series for which the coefficients are palindromic polynomials of a parameter?

More specifically, for a particular function f(x;a) with x, a, and f() in ℝ¹, a > 0, and an exponential power series representation

F₀ + F₁x/1! + F₂x²/2! + F₃x³/3! +...

each of the coefficients, F₁, F₂, F₃, ..., is a palindromic polynomial in a (e.g., F₁ = a + 1, F₂ = 2a² - a + 2, F₃ = 111a³ - 142a² - 142a + 111).

The roots of each polynomial are about equally divided between complex conjugate pairs and real reciprocal pairs. For a polynomial of odd degree, -1 is always a root.
None of the roots of successive polynomials is the same, so I think the series cannot be hypergeometric or q-hypergeometric. The real roots of each of the polynomials are positive (except the single -1 for each odd-degree polynomial), and they appear to converge to an integer value for successive polynomials (e.g., for F₃, F₄, ..., F₁₂,
the largest real root is 1.6862, 2.7565, 3.2367, 3.4961, 3.6493, 3.7457, 3.8096, 3.8850, 3.9079, 3.9251; for a different f(), the roots converge to 9).

Any information about such a power series would be greatly appreciated.

 

[haruspex]

If you substitute b = √a, y = bx and write as a power series in y, the coefficient polynomials take the form Ʃc_n(bⁿ+b^-n), which suggests hyperbolic trig functions.