Counting monomials in skew-symmetric+diagonal matrices
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This question is motivated by Richard Stanley's answer to this MO question. Let $g(n)$ be the number of distinct monomials in the expansion of the determinant of an $ntimes n$ generic " skew-symmetric $+$ diagonal " matrix. For example, $g(3)=4$ since begin{align*} detbegin{pmatrix} x_{1,1}&x_{1,2}&x_{1,3} \ -x_{1,2}&x_{2,2}&x_{2,3} \ -x_{1,3}&-x_{2,3}&x_{3,3} end{pmatrix} &=x_{1, 1}x_{2, 2}x_{3, 3}+x_{1, 1}x_{2, 3}^2+x_{1, 2}^2x_{3, 3} +x_{1, 3}^2x_{2, 2}. end{align*} The sequence $g(n)$ seems to have found a match in OEIS with the generating function $$ sum_{ngeq 0} g(n)frac{x^n}{n!} = frac{e^x}{1-frac12x^2}.$$ QUESTION. Is it true and can you furnish a proof for $$g(n)=sum_{k=0}^{lfloor frac{n}2rfloor}frac{n!}{(n-2k)!,,2^k}?$