The sequence
A059710
starts 1,0,1,1,4,10,35,...
This satisfies the polynomial recurrence relation
$$ (n+5)(n+6)a(n)=2(n-1)(2n+5)a(n-1)+(n-1)(19n+18)a(n-2)+14(n-1)(n-2)a(n-3) $$
I have a $q$-analogue of this sequence. The first few terms are:

$$1$$
$$0$$
$$1$$
$$q^{3}$$
$$q^{6} + q^{4} + q^{2} + 1$$
$$q^{9} + q^{8} + 2 q^{7} + 2 q^{6} + 2 q^{5} + q^{4} + q^{3}$$
$$q^{14} + q^{13} + 4 q^{12} + 2 q^{11} + 5 q^{10} + 4 q^{9} + 5 q^{8} +
2q^{7} + 5 q^{6} + q^{5} + 2 q^{4} + q^{3} + q^{2} + 1$$
$$q^{21} + q^{19} + 2 q^{18} + 4 q^{17} + 5 q^{16} + 9 q^{15} + 10 q^{14}
+ 13 q^{13} + 13 q^{12} + 14 q^{11} + 12 q^{10} + 12 q^{9} + 8 q^{8} + 7
q^{7} + 4 q^{6} + 3 q^{5} + q^{4} + q^{3}$$

These are $q$-analogues since if you put $q=1$ you get the original sequence.

Would anyone like to suggest a $q$-analogue of the polynomial recurrence relation?

I have asked a closely related question in 17610

I can calculate a few more terms than I have posted here.

Since you asked, the polynomial is constructed as follows: take $V$ to be the seven dimensional representation of $G_2$; take the invariant tensors in $\otimes^nV$; take the Frobenius character of this representation of $S(n)$; take the fake degree polynomial of this symmetric function (almost the principal specialisation).

**Further information** In response to Will's comment:

Evaluating at $q=-1$ gives
$$ 1,0,1,-1,4,-2,13,-10,55,-40,241,-190,\ldots $$
Reducing modulo $1+q+q^2$ gives

$$1,0,1,1,1,1,5,3,5,19,15,19,\ldots$$
Reducing modulo $1+q^2$ gives
$$1,0,1,-q,0,0,q,q-1,3,0,2q+3,-q-1,\ldots$$
Reducing modulo $(1-q^5)/(1-q)$ gives
$$1,0,1,q^3,-q^3,0,0,0,0,-q^3-q-1,3,0,\ldots$$
Reducing modulo $1-q+q^2$ gives
$$1,0,1,-1,1,1,1,-1,1,-1,1,-1,\ldots$$

As requested by Jacques, I have put the first fifteen polynomials in a file which you should be able to access here G2 polynomials

I have put the first forty polynomials of a second example in a file which you should be able to access here A1 polynomials These are $q$-analogues of the Riordan numbers The linear recurrence relation is given there as $$ (n+1)*a[n] = (n - 1)*(2*a[n - 1] + 3*a[n - 2]) $$