h[x_]=-30240 + 302400 x^2 - 403200 x^4 + 161280 x^6 - 23040 x^8 + 1024 x^10
(-h''[x]+ 2 x h'[x]+h[x])/h[x]
Simplify[%]
Out[7]= 21
h9[x_]=1/20 h'[x]
Simplify[%]
Expand[%]
                         3          5         7        9
Out[6]= 30240 x - 80640 x  + 48384 x  - 9216 x  + 512 x

h11[x_]=2(x h[x]-10 h9[x])
Simplify[%]
Expand[%]
                             3            5           7          9         11
Out[9]= -665280 x + 2217600 x  - 1774080 x  + 506880 x  - 56320 x  + 2048 x

Exp[-z^2 +2 z x]
Series[%,{z,0,11}]
Simplify[%]
SeriesCoefficient[%,10] 10!
Expand[%]
                          2           4           6          8         10
Out[14]= -30240 + 302400 x  - 403200 x  + 161280 x  - 23040 x  + 1024 x

SeriesCoefficient[%%%,9] 9!
Expand[%]
                          3          5         7        9
Out[16]= 30240 x - 80640 x  + 48384 x  - 9216 x  + 512 x

SeriesCoefficient[%%%%%,11] 11!
Expand[%]
                              3            5           7          9         11
Out[20]= -665280 x + 2217600 x  - 1774080 x  + 506880 x  - 56320 x  + 2048 x

--class4--
The aim of these problems is practice the skill of "integration by operator
methods".  These problems should not be super-hard, but you can test your
algebra by explicit mathematica integration. E.g.:

psi[n_,x_]=HermiteH[n, x] Exp[-x^2/2] /Sqrt[ Sqrt[Pi] 2^n n!]
Integrate[psi[3,x] x^2 psi[3,x],{x,-Infinity,Infinity}]
        7
Out[2]= -
        2
Integrate[psi[3,x] psi[3,x],{x,-Infinity,Infinity}]
Out[3]= 1
Integrate[psi[3,x] psi[4,x],{x,-Infinity,Infinity}]
Out[4]= 0
