fe[n_, x_] := (n + x)^(n + x)/((E n)^(2 x) (n - x)^(n - x))

p[nn_] :=
 Product[(fe[n, 1/2] fe[n, 7/12] fe[n, 1/20] fe[n, 13/20])/(
  fe[n, 1/4] fe[n, 1/12] fe[n, 3/20] fe[n, 11/20]), {n, 1, nn}]

pr = Exp[(47 Catalan)/(30 [Pi]) + 3/4] Sqrt[
   33/(91 [Pi]) Sqrt[2/[Pi] 11^(1/5)/7^(1/3) (3^3/13^3)^(1/5)]];

N[pr] = 0.780459...