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?