degrees of varieties K(2,n)^m / Chebyshev polynomials

[Ralf Stephan]

Hello,
from the empiricist's gumbo:

Consider the sequences

%S A013699 1,32,610,9842,147798,2145600,30664890,435668420,6186432967
%N A013699 Degree of variety K_{2,n}^2.

That seems to be
? tc(n)=if(n==1,1,1/(1-x*tc(n-1)))
? for(n=1,30,t=polcoeff(tc(n),2*n+1):if(1,print1(t",")))
0,1,64,1597,29525,479779,7319744,108126325,1571649221,22682699779,
i.e., the 2n+1-th coefficient in the expansion of the n-th polynomial.

%S A013701 1,512,75025,7174454,562110290,39541748736,2610763825782,
%T A013701 165745451110910,10262482704258873
%N A013701 Degree of variety K_{2,n}^4.

And that seems to be

? for(n=1,30,t=polcoeff(tc(n),3*n+1):if(1,print1(t",")))
0,1,512,75025,7174454,562110290,39541748736,2610763825782,
i.e., the 3n+1-th coefficient in the expansion of the n-th polynomial.

Please either tell me if it's known (I have no easy way to see that here)
or not, so I can either submit more terms or new sequences with the
conjecture.  Or generalize and prove it, it's all up to you.

Many thanks for your comment,
ralf