If we weight instead by the Tutte polynomial. The case of spanning trees

>    Qtser:=proc(n)
subs(q=(mu-1)*(nu-1), w=w/(nu-1), Qser(n)): end:

>    map(normal,Qtser(3));

1+t*mu*w*y^2+y*w*(z*nu*mu+y*z*x*nu+2*w*y^3*mu^2+y*z*x*mu)*t^2+y*w*(2*z^2*x*nu^2*mu+z^2*x*nu^2+y*x^2*nu^2*z^2+3*y^2*z*w*mu^2*nu+z^2*x*nu*mu+3*w*y^3*z*x*mu*nu+y^2*z*w*nu+z^2*x^2*y*nu+3*w*y^3*z*x*mu^2+z^2...
1+t*mu*w*y^2+y*w*(z*nu*mu+y*z*x*nu+2*w*y^3*mu^2+y*z*x*mu)*t^2+y*w*(2*z^2*x*nu^2*mu+z^2*x*nu^2+y*x^2*nu^2*z^2+3*y^2*z*w*mu^2*nu+z^2*x*nu*mu+3*w*y^3*z*x*mu*nu+y^2*z*w*nu+z^2*x^2*y*nu+3*w*y^3*z*x*mu^2+z^2...

The case mu=nu=1 (spanning trees, cf beginning of section 11). We only count quasi-triangulations (no internal digon, x=0) with outer degree 2

>    m:=12:coeff(subs(mu=1, nu=1, z=1,x=0,t=1,map(normal,Qtser(m))),y,2);

w+10*w^2+210*w^3+6006*w^4

>    seq(binomial(2*n,n)*binomial(4*n-2,2*n-1)/2/n/(n+1),n=1..6);

1, 10, 210, 6006, 204204, 7759752

>    unassign('n'):simplify(normal(convert(asympt(binomial(2*n,n)*binomial(4*n-2,2*n-1)/2/n/(n+1),n,4),polynom)));

1/16/Pi*2^(1/2)/n^3*64^n

>