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Find Generating Functions for Sequences
Find a generating function whose power series coefficients are the elements in a list.
In[1]:=

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data = {1, 2/5, 1/10, 2/105, 1/336, 1/2520, 1/21600, 1/207900, 1/

   2217600, 1/25945920};
In[2]:=

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FindGeneratingFunction[data, x] // Simplify
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In[3]:=

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FindGeneratingFunction[data, x] // Simplify;

CoefficientList[Series[%, {x, 0, 9}], x]
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In[4]:=

Click for copyable input
data = {1, 2/5, 1/10, 2/105, 1/336, 1/2520, 1/21600, 1/207900, 1/

   2217600, 1/25945920};;

CoefficientList[Series[%, {x, 0, 9}], x];

% - data
Out[4]=