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a guest Jun 27th, 2019 68 Never
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  1. ClearAll[p, b];
  2. $Assumptions = {b > 0};
  3. p[m_, n_] := p[n, m] = If[ m < 1 || n < 1 || n > m + 1, 0 # &,
  4.             Module[{s}, Integrate[Exp[(# - s) b n]
  5.                 (KroneckerDelta[m, 1] KroneckerDelta[n, 2] b  
  6.                  + b n p[m, n + 1][s] + If[m < 2, 0,
  7.                    b n (n - 1) (p[m - 1, n - 2][s] + p[m - 1, n - 1][s]
  8.                                 +p[m - 1, n][s])]),
  9.                 {s, 0, #}] &]];
  10.      
  11. In[5]:= p[5, 6][t] // AbsoluteTiming
  12.  
  13. Out[5]= {83.3974,
  14.  15 + 86 E^(b t) - 47350 E^(5 b t) + 5034 E^(6 b t) +
  15.   3240 E^(3 b t) (34 + 15 b t) + 10 E^(2 b t) (361 + 30 b t) +
  16.   5 E^(4 b t) (-14311 + 21216 b t)}
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