A 3rd formula for the central Delannoy numbers? There are several in the literature proving the two alternative formulas for the (diagonal) Delannoy numbers; namely that
$$d_n=\sum_{k=0}^n\binom{n}k\binom{n+k}k=\sum_{k=0}^n\binom{n}k^22^k.$$
Each formulation might have an advantage over the other, depending on the context.
I have not seen (still willing to be referred to) the below bizarre-looking expression . So, I ask:

QUESTION. Is there a combinatorial justification for this?
$$d_n=\sum_{k=\lfloor\frac{n}2\rfloor}^n(-1)^{n+k}\binom{2k}k\binom{k}{n-k}3^{2k-n}2^{-n}.$$

The motivation comes from studying divisibility questions and a direct look at the generating function
$$\frac1{\sqrt{1-6x+x^2}}.$$
 A: By convolution,
$$ D_n=\sum_{k=0}^{n}\binom{n}{k}^2 2^k = \sum_{k=0}^{n}\binom{n}{k}2^k\binom{n}{n-k}= [z^n](1+2z)^n(1+z)^n = [z^n](1+3z+2z^2)^n $$
hence by Cauchy's integral formula
$$ D_n = \frac{1}{2\pi i}\oint_{|z|=\varepsilon}(1+3z+2z^2)^n \frac{dz}{z^{n+1}} $$
where the only singularity of the integrand function is at the origin, so the value of the integral is unaffected if we consider $|z|=\frac{1}{\sqrt{2}}$ as a new integration path. A change of variable leads to
$$ D_n = \frac{1}{2\pi}\int_{0}^{2\pi}(3+2\sqrt{2}\cos\theta)^n\,d\theta $$
from which it is simple to derive the asymptotic behaviour of $D_n$ (through Laplace's method). Additionally, by the binomial theorem
$$ D_n = \sum_{k=0}^{n}\binom{n}{k}8^{k/2}3^{n-k}\cdot\frac{1}{2\pi}\int_{0}^{2\pi}\left(\cos\theta\right)^k\,d\theta $$
which simplifies into
$$ D_n = \sum_{j=0}^{\lfloor n/2\rfloor} \binom{n}{2j}8^j 3^{n-2j}\cdot\frac{1}{4^j}\binom{2j}{j}= \sum_{j=0}^{\lfloor n/2\rfloor}\binom{n}{2j}\binom{2j}{j}2^j 3^{n-2j}=\sum_{j=0}^{\lfloor n/2\rfloor}\binom{n}{j}\binom{n-j}{j} 2^j 3^{n-2j}$$
that is equivalent to the claim.
