We have
$$
S(k,x) = \sum\limits_{n = 0}^\infty {\left( \matrix{ 2n + k \cr n \cr} \right)x^{2n + k} }
= x^k \sum\limits_{n = 0}^\infty {\left( \matrix{ 2n + k \cr n \cr} \right)\left( {x^2 } \right)^n }
$$
Indicating the series coefficients as
$$
t_n = \left( \matrix{ 2n + k \cr n \cr} \right)
$$
then we have
$$
\eqalign{
& t_0 = 1 \cr & {{t_{n + 1} } \over {t_n }} =
{{\left( \matrix{ 2n + 2 + k \cr n + 1 \cr} \right)} \over {\left( \matrix{ 2n + k \cr n \cr} \right)}} =
{{n!\left( {n + k} \right)!\left( {2n + 2 + k} \right)!} \over {\left( {n + 1} \right)!\left( {n + 1 + k} \right)!\left( {2n + k} \right)!}}
= {{\left( {2n + 2 + k} \right)\left( {2n + 1 + k} \right)} \over {\left( {n + 1} \right)\left( {n + 1 + k} \right)}} = \cr
& = 4{{\left( {n + 1 + k/2} \right)\left( {n + 1/2 + k/2} \right)} \over {\left( {n + 1} \right)\left( {n + 1 + k} \right)}} \cr}
$$
which means that the series might be expressed through a
Hypergeometric function
$$
\eqalign{
& S(k,x) = x^k \sum\limits_{0\, \le \;k} {{{\left( {1/2 + k/2} \right)^{\,\overline {\,n\,} }
\left( {1 + k/2} \right)^{\,\overline {\,n\,} } } \over {\left( {1 + k} \right)^{\,\overline {\,n\,} } }}
{{\left( {4x^2 } \right)^{\,n} } \over {n!}}} = \cr
& = x^k {}_2F_{\,1} \left( {\left. {\matrix{ {1/2 + k/2,\;1 + k/2} \cr {1 + k} \cr } \;}
\right|\;4x^2 } \right) \cr}
$$
which converges for $|x|<1/2$
To explain the convergence range, it is known that
$$
S(0,x) = \sum\limits_{n = 0}^\infty {\left( \matrix{ 2n \cr n \cr} \right)x^{2n} } =
{1 \over {\sqrt {1 - 4x^2 } }}
$$
The duplication formula for Gamma in fact gives
$$
\eqalign{
& \left( \matrix{
2n \cr
n \cr} \right) = {{\Gamma \left( {2n + 1} \right)} \over {\Gamma \left( {n + 1} \right)^2 }} =
{{4^{\,n} } \over {\sqrt \pi }}{{\Gamma \left( {n + 1/2} \right)} \over {\Gamma \left( {n + 1} \right)}} = \cr
& = 4^{\,n} {{\Gamma \left( {n + 1/2} \right)} \over {\Gamma \left( {1/2} \right)\Gamma \left( {n + 1} \right)}} =
4^{\,n} \left( \matrix{ n - 1/2 \cr n \cr} \right) = \cr
& = \left( { - 4} \right)^{\,n} \left( \matrix{ - 1/2 \cr n \cr} \right) \cr}
$$