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I have a following sequence:

 Table[Sum[Sum[(1 + j + k + n)!/((1 + j + k) j! k! ), {j, 0, n}], {k, 0, n}], {n, 1, 5}]
 (* {16, 542, 31488, 2646024, 292224000} *)

The sum inside can be evaluated:

 Sum[(1 + j + k + n)!/((1 + j + k) j! k! ), {j, 0, n}]
 (* -(((2 + k + 2 n)! HypergeometricPFQ[{1, 2 + k + n, 3 + k + 2 n}, {2 + n, 3 + k + n}, 1])/((2 + k + n) k! (1 + n)!)) *)

But the same sum with substitution gives an error message.

 Table[Sum[-(((2 + k + 2 n)! HypergeometricPFQ[{1, 2 + k + n, 3 + k + 2 n}, {2 + n, 3 + k + n}, 1])/((2 + k + n) k! (1 + n)!)), {k, 0, n}], {n, 1, 5}]
 (* Infinity::indet: Indeterminate expression ComplexInfinity+ComplexInfinity encountered. *)

Why aren't the results the same?

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  • $\begingroup$ Are you aware that you don't need to use Sum[] twice? Sum[(1 + j + k + n)!/((1 + j + k) j! k!), {j, 0, n}, {k, 0, n}] $\endgroup$ – J. M. will be back soon Apr 3 at 12:59
  • $\begingroup$ Yes, of course (see tens of my programs in the OEIS, for example in oeis.org/A144661) $\endgroup$ – Vaclav Kotesovec Apr 3 at 13:07
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The problem that the hypergeometric result returned is not good for integer arguments:

Table[HypergeometricPFQ[{1, 2 + k + n, 3 + k + 2 n}, {2 + n, 3 + k + n}, 1],
      {n, 2}, {k, 0, n}]
   {{ComplexInfinity, ComplexInfinity}, {ComplexInfinity, ComplexInfinity, ComplexInfinity}}

which spoils further computations.

A useful tactic is to flip indices in the inner sum:

Sum[(1 + n - j + k + n)!/((1 + n - j + k) (n - j)! k!), {j, n, 0, -1}]
   ((1 + k + 2 n)! HypergeometricPFQ[{1, -1 - k - n, -n}, {-1 - k - 2 n, -k - n}, 1])/
   ((1 + k + n) k! n!)

which should now be usable:

Table[Sum[((1 + k + 2 n)! HypergeometricPFQ[{1, -1 - k - n, -n},
          {-1 - k - 2 n, -k - n}, 1])/((1 + k + n) k! n!), {k, 0, n}], {n, 1, 5}]
   {16, 542, 31488, 2646024, 292224000}

It should be noted that index flipping does not always give a useful result, but we are fortunate in this case.

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  • $\begingroup$ Yes, this is a good idea, thank you! $\endgroup$ – Vaclav Kotesovec Apr 3 at 13:22

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