# 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?

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 $$varepsilon > 0$$.
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.