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I presume you want the coefficients $$a_k=\int_{-\infty}^\infty p(M)\psi_k(M)\,dM,$$ with $$\psi_k(x)=\frac{1}{\pi^{1/4}\sqrt{ 2^k k!}} e^{-x^2/2} H_k(x)$$ the normalized Hermite function of order $k$ and $p(M)$ the probability density function of the median $M$ of the $n$ Gaussian random variables.

For $n\gg 1$ the median of $n$ i.i.d. normal random variables has a Gaussian distribution $p(M)=(2\pi\sigma^2)^{-1/2}e^{-M^2/2\sigma^2}$, with $\sigma^2=\pi/2n$, resulting in $a_{2p+1}=0$, $$a_{2p}=\frac{(2p)!}{2^{p}\pi^{1/4}p!\sqrt{(2p)!}}\frac{(\sigma^2-1)^p}{(1+\sigma^2)^{p+1/2}}.$$

I presume you want the coefficients $$a_k=\int_{-\infty}^\infty p(M)\psi_k(M)\,dM,$$ with $$\psi_k(x)=\frac{1}{\pi^{1/4}\sqrt{ 2^k k!}} e^{-x^2/2} H_k(x)$$ the normalized Hermite function of order $k$ and $p(M)$ the probability density function of the median $M$ of the $n$ Gaussian random variables.

For $n\gg 1$ the median of $n$ i.i.d. normal random variables has a Gaussian distribution $p(M)=(2\pi\sigma^2)^{-1/2}e^{-M^2/2\sigma^2}$, with $\sigma^2=\tfrac{1}{2}\pi/n$, resulting in $a_{2m+1}=0$, $$a_{2m}=\frac{(-1)^m\sqrt{(2m)!}}{2^{m}\pi^{1/4}m!}\frac{\bigl(1-\tfrac{1}{2}\pi/n\bigr)^m}{\bigl(1+\tfrac{1}{2}\pi/n\bigr)^{m+1/2}}.$$

I presume you want the coefficients $$a_k=\int_{-\infty}^\infty p(M)\psi_k(M)\,dM,$$ with $$\psi_k(x)=\frac{1}{\pi^{1//4}\sqrt{ 2^k k!}} e^{-x^2/2} H_k(x)$$ the normalized Hermite function of order $k$ and $p(M)$ the probability density function of the median $M$ of the $n$ Gaussian random variables.

For $n\gg 1$ the median of $n$ i.i.d. normal random variables has a Gaussian distribution $p(M)=(2\pi\sigma^2)^{-1/2}e^{-M^2/2\sigma^2}$, with $\sigma^2=\pi/2n$, resulting in $a_{2p+1}=0$, $$a_{2p}=\frac{(\sigma^2-1)^p}{\pi^{1/4}2^{p-1}(1+\sigma^2)^{p+1/2}}.$$ (I miss a numerical coefficient, will insert that soon)

I presume you want the coefficients $$a_k=\int_{-\infty}^\infty p(M)\psi_k(M)\,dM,$$ with $$\psi_k(x)=\frac{1}{\pi^{1/4}\sqrt{ 2^k k!}} e^{-x^2/2} H_k(x)$$ the normalized Hermite function of order $k$ and $p(M)$ the probability density function of the median $M$ of the $n$ Gaussian random variables.

For $n\gg 1$ the median of $n$ i.i.d. normal random variables has a Gaussian distribution $p(M)=(2\pi\sigma^2)^{-1/2}e^{-M^2/2\sigma^2}$, with $\sigma^2=\pi/2n$, resulting in $a_{2p+1}=0$, $$a_{2p}=\frac{(2p)!}{2^{p}\pi^{1/4}p!\sqrt{(2p)!}}\frac{(\sigma^2-1)^p}{(1+\sigma^2)^{p+1/2}}.$$

I presume you want the coefficients $$a_k=\int_{-\infty}^\infty p(M)\psi_k(M)\,dM,$$ with $$\psi_k(x)=\frac{1}{\pi^{1//4}\sqrt{ 2^k k!}} e^{-x^2/2} H_k(x)$$ the normalized Hermite function of order $k$ and $p(M)$ the probability density function of the median $M$ of the $n$ Gaussian random variables.

For $n\gg 1$ the median of $n$ i.i.d. normal random variables has a Gaussian distribution $p(M)=(2\pi\sigma^2)^{-1/2}e^{-M^2/2\sigma^2}$, with $\sigma^2=\pi/2n$, resulting in $a_{2p+1}=0$, $$a_{2p}=\frac{(\sigma^2-1)^p}{\pi^{1/4}2^{p-1}(1+\sigma^2)^{p+1/2}}.$$

I presume you want the coefficients $$a_k=\int_{-\infty}^\infty p(M)\psi_k(M)\,dM,$$ with $$\psi_k(x)=\frac{1}{\pi^{1//4}\sqrt{ 2^k k!}} e^{-x^2/2} H_k(x)$$ the normalized Hermite function of order $k$ and $p(M)$ the probability density function of the median $M$ of the $n$ Gaussian random variables.

For $n\gg 1$ the median of $n$ i.i.d. normal random variables has a Gaussian distribution $p(M)=(2\pi\sigma^2)^{-1/2}e^{-M^2/2\sigma^2}$, with $\sigma^2=\pi/2n$, resulting in $a_{2p+1}=0$, $$a_{2p}=\frac{(\sigma^2-1)^p}{\pi^{1/4}2^{p-1}(1+\sigma^2)^{p+1/2}}.$$ (I miss a numerical coefficient, will insert that soon)