A power of a sum in a non-commutative algebra

Let $ A := mathbb{C}langle x, y rangle / ( xy-2 ) $ where $ mathbb{C}langle x, y rangle $ is the free (non-commuative) $ mathbb{C} $-algebra which is generated by $ x $ and $ y $ and $ ( xy-2 ) $ is the two-sided ideal which generated by the element $ xy-2 $.

Let us also write $ x $ resp. $ y $ for the classes of $ x $ resp. $ y $ in $ A $. Clearly, the elements $ y^k x^l $ form a basis of $ A $ for $ k, l in mathbb{N}_0 $. Therefore, there is a presentation
$$ ( x + y )^n = sum_{k + l = 0, dots, n} c_{k,l}^{(n)} y^k x^l $$
for any $ n in mathbb N_0 $.

Questions:
I am interested in the sum of these coefficients, i.e.
$$ S_n = sum_{k, l} c_{k,l}^{(n)}. $$

  1. Is there a general method to compute powers of sums in such algebras?
  2. What can be more said about the asymptotic behaviour of $ S_n $ except for the obvious bounds in $(*)$?
  3. Does anybody know a context in which this algebra (or more generally the algebra $ mathbb Clangle x, y rangle / ( xy – a ) $ for $ a in mathbb N $) plays a role?

Known and maybe helpful:
As we have two choices in each of the $ n $ factors in $ ( x + y )^n $, (without using the relation $ xy = 2 $) we already count at least $ 2^n $ $ ( x, y ) $-monomials. Moreover, applying the relation $ xy = 2 $ only doubles the coefficient of any of these $ 2^n $ monomials. This provides the lower bound $ 2^n $ for $ S_n $.
One the other hand, mapping $ ( x, y ) mapsto (sqrt 2, sqrt 2) $ yields a morphism $ A to mathbb{C} $ of $ mathbb C $-algebras and gives the upper bound $ 2^{3n/2} $. So, we have the bounds
begin{align} 2^n le S_n le 2^{3n/2}. (*) end{align}
Furthermore, with some more effort one can show $ S_n ll 2^{3n/2} / sqrt n $ asymptotically as $ n to infty $. Anyway, I have the feeling that $ 2^{3n/2} $ is not far off, in the sense that $ (2^{3/2}-varepsilon)^n / S_n to 0 $ for any $ varepsilon > 0 $.

Approach:
This algebra $ A $ is isomorphic to the sub algebra $ B $ of all $ mathbb C $-linear operators on the vector space $ mathbb C^{mathbb N } $ of sequences with elements in $ mathbb C $ which is generated by the forward shift $ R $ and backward shift $ L $ (consider the map $ ( x, y ) mapsto ( 2 L, R ) $). I seems to be possible to get $ S_n $ by the action of $ B $ on $ mathbb C^{mathbb N } $. But it is computationally involved and unclear whether the so obtained presentation of the $ S_n $ is useful.