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I need to calculate a limit of a sequence given by its recurrence relation. I tried the following:

Limit[DifferenceRoot[Function[{y, n}, {
      - n (1 + n)^2 (5 + 2 n) y[n] 
      - (1 + n) (3 + n) (11 + 14 n + 4 n^2) y[1 + n] 
      + (-77 - 128 n - 69 n^2 - 12 n^3) y[2 + n] 
      + (29 + 84 n + 79 n^2 + 30 n^3 + 4 n^4) y[3 + n] 
      + (3 + n)^3 (3 + 2 n) y[4 + n] == 0, 
      y[1] == 0, y[2] == -1, y[3] == 1/8, y[4] == -215/216
  }]][n], n -> Infinity]

but it returned unevaluated.

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First, you can try to apply the FunctionExpand command to the DifferenceRoot object. If it is able to find a closed form of the sequence, then the Limit might be able to find an exact symbolic limit.

To find a numerical approximation, you can use the SequenceLimit command. In general, it does not guarantee to give the correct result, but if your sequence behaves 'nicely' then you might be able to obtain the correct result with a very high precision:

In[1]:= SequenceLimit[N[Table[
        DifferenceRoot[Function[{y, n}, {
            - n (1 + n)^2 (5 + 2 n) y[n] 
            - (1 + n) (3 + n) (11 + 14 n + 4 n^2) y[1 + n] 
            + (-77 - 128 n - 69 n^2 - 12 n^3) y[2 + n] 
            + (29 + 84 n + 79 n^2 + 30 n^3 + 4 n^4) y[3 + n] 
            + (3 + n)^3 (3 + 2 n) y[4 + n] == 0, 
            y[1] == 0, y[2] == -1, y[3] == 1/8, y[4] == -215/216
        }]][n],
    {n, 1, 500}], 200]]

Out[1]= -0.442460189377912495218798219174656335184133627022583585864263293471236392630861

Plugging the numeric result in the TranscendentalRecognize function with a wide enough set of transcendental constants, you might get its possible closed form:

-1/12 (-π^2 Log[2] + 4 Log[2]^3 + 9 Zeta[3])
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