even later I did not say that it would give you what you wanted, only that I thought it might. Taking up my idea for powers of 2 gives two sequences of polynomials $1,2s+2,4s^2+10s+2,8s^3+36s^2+18s+2,\dots$ and $2,6s+2,16s^2+14s+2,40s^3+64s^2+22s+2,\dots$. Sending the first to $1,0,0,0,\dots$ sends the powers of $s$ to $1,-1,2,-7,38,-295,3098,\dots$ which look to be the absolute values of Genocchi numbers of second kind divided by 2^(n-1). This sends the other sequence to $2,-4,20,-172,2228,-40300,\dots$. The OEIS does not immediately identify that. Dropping the first term, still nothing. Since the remaining terms are multiples of 4 one divides that out to get $-1, 5, -43, 557, -10075, 241949, -7437547,\dots$ which does get identified as values B(2,n)/4 of Gandhi polynomials along with a reference to an article on combinatorial interpretations of Genochi numbers. I did not identify anything happening if we try to send the second sequence to $2,0,0,0,\dots$. I did not try for a functional equation.

later: The OEIS and associated Journal of Integer Sequences are full of sequences and transformations taking one to another. The example above was an illustration (the only one I tried) of what I am sure would generate many examples (probably some nicer than the one I gave). Start with an integer recurrence of the form $a_0=0$ $a_1=1$ and $a_m=u(m)a_{m-1}+v(m)a_{m-2}$. Replace this with $A_0=1$ $A_1=1$ and $A_m=u(m)a_{m-1}+sv(m)a_{m-2}$. Then (in general) $A_{2t}$ and $A_{2t+1}$ are both polynomials of degree $t$ so one has two bases for the additive vector-space of polynomials. Take the linear transformation which sends one to $1,0,0,0,\dots$ and see what it does to the other (or some other basis of that space). Look up the associated sequence (perhaps the numerators and denominators if it is a rational sequence) in the OEIS and see what you got. Examples (which I have not investigated) include powers of 2 ($a_n=a_{n-1}+2a_{n-2}$) and factorials ($a_n=(n-1)a_{n-1}+(n-1) a_{n-2}$). Those two are shifted, it would be nicer to start with $a_0=1$ and $a_1=1$ making appropriate adjustments. Starting the second example with $a_0=1$ and $a_1=0$ gives derangements. One could consider other initial conditions $A_0=c_0$ and $A_1=c_1$

There are many sequences of orthogonal polynomials like the Chebshev polynomials I mentioned above, (Legendre, Laguerre, Hermite, Jacobi,etc.) which are alternately even and odd. Dividing the odd ones by $x$ and then replacing $x$ with $\sqrt{x}$ (maybe taking absolute value of coefficients) gives more examples. Exponential generating functions might be more appropriate in some cases.

All that we are using of the polynomial structure is the (additive) book-keeping for the coordinates of a vector-space. Of course for famous sequences of polynomials that is still interesting. In general we could instead look at infinite lower triangular matrices. Then I think that your linear transformations correspond to the first column of the inverse matrix and the image on the companion sequence is simply the first column of a certain matrix product. Other columns might be of interest as well.