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I have being calculate some Series Coefficient and the result was and DifferenceRoot equation but i do not how

(-1)^v 2^(1 + v) 3^(-1 - v) Binomial[n + λ, n] Gamma[
  1 + v] DifferenceRoot[
   Function[{\[FormalY], \[FormalN]}, {3 \[FormalN] (\[FormalN] - 
λ) \[FormalY][\[FormalN]] + (-6 - 13 \[FormalN] - 
          10 \[FormalN]^2 + 2 \[FormalN] n + 6 λ + 
          9 \[FormalN] λ) \[FormalY][
         1 + \[FormalN]] + (22 + 31 \[FormalN] + 12 \[FormalN]^2 - 
          4 n - 4 \[FormalN] n - 12 λ - 
          9 \[FormalN] λ) \[FormalY][
         2 + \[FormalN]] + (2 + \[FormalN]) (-11 - 6 \[FormalN] + 
          2 n + 3 λ) \[FormalY][
         3 + \[FormalN]] + (2 + \[FormalN]) (3 + \[FormalN]) 
\[FormalY][4 + \[FormalN]] == 0, \[FormalY][-1] == 0, \[FormalY][0] ==
       0, \[FormalY][1] == 
      Hypergeometric2F1[1, -n, 1 + λ, 2/3], \[FormalY][2] == (
      3 Hypergeometric2F1[1, -n, 1 + λ, 2/3] + 
       3 λ Hypergeometric2F1[1, -n, 1 + λ, 2/3] - 
       2 n Hypergeometric2F1[2, 1 - n, 2 + λ, 2/3])/(3 + 3 λ)}]][1 + v]

It is possible extraction as normal equation to get the coefficient of the series I have the feeling that Mathematica make a great job but the result is it not easy use to application; thanks anyway.

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  • $\begingroup$ What is your question? $\endgroup$ – MarcoB Apr 28 '18 at 22:29
  • $\begingroup$ how extraction the coefficient of the DifferenceRoot $\endgroup$ – capea Apr 28 '18 at 22:34
  • $\begingroup$ Thanks anyway MarcoB for your help i have been reading the Mathematica help center about the DifferentRoot but it is not very clear for me $\endgroup$ – capea Apr 28 '18 at 22:43
  • $\begingroup$ Your code returns an error: DifferenceRoot::icond: Initial conditions should be of the form y[a] == b0, y[a + 1] == b1, ... please fix this so that it can be run. $\endgroup$ – bill s Apr 29 '18 at 1:23
  • 1
    $\begingroup$ Assign values to n and v: Table[expr, {n, 0, 3}, {v, 1, 3}] // FullSimplify $\endgroup$ – Bob Hanlon Apr 29 '18 at 2:55

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