Could Maple confirm below Mathematica result (desirably without using the trick of adding 10^-20 to the values of "n") ?
.
 W[n_] := 1/(3 n Sqrt[Pi] Gamma[2 - n] Gamma[2 + n])*2^(-7 - 4 n) (3 256^
  n Gamma[2 - n] Gamma[-(1/2) + 
    n] (8 n (-3 + 5 n + 2 n^2) Hypergeometric2F1[1 - n, 
      2 - n, -2 n, 3/
      4] - (-1 + 
       n) (4 (-4 + 7 n + 2 n^2) Hypergeometric2F1[2 - n, 2 - n, 
         1 - 2 n, 3/4] - (-10 + 3 n + n^2) Hypergeometric2F1[
         2 - n, 3 - n, 2 - 2 n, 3/4])) - 
 9^n Gamma[-(1/2) - n] Gamma[
   2 + n] (8 n (5 + 11 n + 2 n^2) Hypergeometric2F1[1 + n, 2 + n, 
      2 n, 3/4] - 
    3 (1 + n) (4 (4 + 9 n + 2 n^2) Hypergeometric2F1[2 + n, 2 + n,
          1 + 2 n, 3/4] - 
       3 (6 + 5 n + n^2) Hypergeometric2F1[2 + n, 3 + n, 
         2 (1 + n), 3/4])));
 Table[N[W[n + 10^-20], 20], {n, 1, 15}] // Rationalize

 (*{2, 14, 106, 838, 6802, 56190, 470010, 3968310, 33747490, 288654574, \
 2480593546, 21400729382, 185239360178, 1607913963614, 13991107041306}*)

Invoked by the OEIS superseeker, Maple "gfun" package "listtoalgeq" identified possible lgdegf for https://oeis.org/A035001

1, 2, 4, 5, 8, 12, 14, 16, 28, 32, 37, 64, 94, 106, 128, 144, 232, 256, 289, 320, 512, 560, 704, 760, 838, 1024, 1328, 1536, 1944, 2048, 2329, 3104, 3328, 4096, 4864, 6266, 6802, 7168, 8192, 11952, 15360, 16384, 16428, 19149, 28928, 32768, 37120, 42168 

as follows:

1024-5120*a(n)+11520*a(n)^2-15360*a(n)^3+13440*a(n)^4-8064*a(n)^5+3360*a(n)^6-960*a(n)^7+180*a(n)^8-20*a(n)^9+a(n)^{10}

The coefficients of above polynomial are:

{1, -20, 180, -960, 3360, -8064,13440, -15360, 11520, -5120, 1024,...}

It is interesting that the absolute values of above polynomial coefficients satisfy a(n) of

https://oeis.org/A013609

for n=55...65,

which is the 11th row in the triangle presentation of A013609, so in other words the absolute values of above polynomial coefficients are T={11, k} for k=1...11

Hi,

Below is the parametric identity, which i have found empirically: sqrt[pi] =(1/(2^j) ((k*gamma[5 + 2 j] gamma[ 1 + l] hypergeometricpfq[{1, 5/2 + j, 3 + j}, {3 + j + l/2, 7/2 + j + l/2}, -1])/ gamma[6 + 2 j + l] + ((k + m) gamma[7 + 2 j] gamma[ 1 + l] hypergeometricpfq[{1, 7/2 + j, 4 + j}, {4 + j + l/2, 9/2 + j + l/2}, -1])/gamma[8 + 2 j + l]))/(2^(-5 - 3 j - l) gamma[ 5 + 2 j] gamma[ 1 + l] (k hypergeometricpfqregularized[{1, 5/2 + j, 3 + j}, {3 + j + l/2, 7/2 + j + l/2}, -1] + 1/2 (3 + j) (5 + 2 j) (k + m) hypergeometricpfqregularized[{1, 7/2 + j, 4 + j}, {4 + j + l/2, 9/2 + j + l/2}, -1]

It seems to be true for arbitrary j,k,l,m parameters where j, k, l and m are signed integer

For all tried specific sets of {j,k,l,m} above identity was confirmed by both mathematica based wolframalpha and maple.

Could this identity be simplified?

How this identity could be proven either by using mathematica and/or analyticall

Thanks,

Best regards,

Alexander R. Povolotsky 

Does the below infinite sum involving primes converge?sum(1/(Prime(n)Prime(n+1)Prime(n+2)*Prime(n+3)),n=1... infinity) Free WolframAlpha app runs out of time while evaluating this sum... The term values which free WolframAlpha app shows for this sum (starting from the term 41 and on) is 0.0059292... If 0.0059292 is indeed the value for this sum, could then more significant digits be produced? Thanks, Best Regards, Alexander R. Povolotsky