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

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data = {{1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12}, {1, 1, 1, 2, 2, 3, 4, 

    5, 6, 8, 10, 12, 15}, {1, 2, 6, 20, 70, 252, 924, 3432, 12870},

   {1, 3/2, 9/5, 2, 15/7, 9/4, 7/3, 12/5}, {1 + a, 1 + a^2, 1 + a^3, 

    1 + a^4, 1 + a^5}, {0, 1, 3, 12, 55, 273, 1428, 7752, 43263, 

    246675, 1430715}};
In[2]:=

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Text@TraditionalForm@

  Grid[{#, FindGeneratingFunction[#, x]} & /@ data, 

   Alignment -> {Left}, Background -> LightYellow, Frame -> All, 

   Spacings -> {1, 2}]
Out[2]=