Consider the Fibonacci polynomials defined by $$F_n(s)=F_{n-1}(s)+sF_{n-2}(s)$$ with initial values $F_0(s)=0$ and $F_1(s)=1$ and define a linear functional  $L$ on the polynomials in $s$ by $$L(F_{2n})=\delta_{n,1}.$$  Then $$L(F_{2n+1})=(2n+1)B_n,$$ where  $B_n$ are the Bernoulli numbers defined by $B_n={\sum{n\choose k} B_k\}$ for $n\ge2$ and $B_0=1.$  
Choosing the linear functional $M$ defined by $$M(F_{2n+1})=\delta_{n,0},$$  gives $$M(F_{2n})=(-1)^n G_{2n},$$  where $G_{2n}$ are the Genocchi numbers $G_{2n}=(-1)^n 2 (1-4^n) B_{2n}.$

Finally let  $H_n$ be a variant of the Hermite polynomials defined by $$H_n(s)=H_{n-1}(s)-(n-1)s H_{n-2}(s)$$  and the linear functional $N$ defined by $$N(H_{2n})=\delta_{n,0},$$ then we get $$N(H_{2n-1})=(-1)^{n-1} T_{2n-1},$$ where $T_{2n-1}=(-1)^{n-1} \frac{4^n (4^n-1)}{2n} B_{2n}$ are the tangent numbers.

 My question is: Are these isolated results or special cases of a more general theorem? Does anyone know other such examples?

**Edit.** To make my question somewhat more precise: Define the Fibonacci polynomials by $F_n(x,s)=xF_{n-1}(x,s)+sF_{n-2}(x,s)$  and the Hermite polynomials by  $H_n(x,s)=xH_{n-1}(x,s)-(n-1)s H_{n-2}(x,s).$
The above results follow from the identities
$$(e^{xz}  + 1)\sum {\frac{{F_{2n} (x,s)}}{{(2n)!}}z^{2n}  =(e^{x z}-1) \sum {\frac{{F_{2n + 1} (x,s)}}{{(2n + 1)!}}z^{2n + 1} } } $$ and
$$(e^{2xz}  + 1)\sum {\frac{{H_{2n + 1} (x,s)}}{{(2n + 1)!}}z^{2n + 1}  = (e^{2xz}  - 1)\sum {\frac{{H_{2n} (x,s)}}{{(2n)!}}z^{2n} } }. $$

Thus a more precise question would be: Are there polynomial sequences which satisfy similar identities?

**Further edit.** A more precise question:  Are there  "naturally occurring sequences" $A_n(x,s)$ satisfying $A_n(x,s)=xA_{n-1}(x,s)+c(n,s)A_{n-2}(x,s)$ such that 
$$\sum {\frac{{A_{2n} (x,s)}}{{(2n)!}}z^{2n}  =b(z,x) \sum {\frac{{A_{2n + 1} (x,s)}}{{(2n + 1)!}}z^{2n + 1} } }, $$  where $b(z,x)$ does not depend on $s?$
The only examples I know besides the Fibonacci and Hermite polynomials are the Lucas polynomials $L_n(x,s)$ defined by $L_n(x,s)=xL_{n-1}(x,s)+sL_{n-2}(x,s)$ with  initial values $L_0(x,s)=2$ and $L_1(x,s)=x.$