Euler product

[old <ramare@agat.univ-lille1.fr>]

Dear all,
 Here is the constants I want to compute:

C(h)=prodeuler(p=2,infinity, (1+(1+1/p)^h/(p-1))/(1+(1+1/p)^(h-1)/(p-1)))

and with h quite large (up to 50).

I have a prodeulerrat by Henri Cohen, but the results are incoherent
from h=19-21 onwards. I tried to compute C(h)/C(h-1) directly but
failed also. Log is below.

Any help appreciated !
Best,
Amities,
                  Olivier

{frel(h)=prodeulerrat((1+(p+1)^h/p^h/(p-1))/(1+(p+1)^(h-1)/p^(h-1)/(p-1)),2)}

{g(n)=local(res,resrel,h); res=1.519817754635066571658191948;
           for(h=2,n,resrel=frel(h);res*=resrel;print(h," -> 
",resrel,"  --> ", res));}

? g(50)
2 -> 1.628355734896019122  --> 2.474803956756801499
3 -> 1.739938267564662721  --> 4.306006109081601676
4 -> 1.849636170351067217  --> 7.964544649109993522
5 -> 1.953806899433914367  --> 15.56118228628056991
6 -> 2.050378439566114165  --> 31.90631265394781175
7 -> 2.138609052758110336  --> 68.23512908186343916
8 -> 2.218655987964094462  --> 151.3902777269792425
9 -> 2.291176534998999884  --> 346.8618519550365689
10 -> 2.357038308463183063  --> 817.5666728025064213
11 -> 2.417137113232311338  --> 1976.170747372795998
12 -> 2.472297452123113924  --> 4885.681903689993376
13 -> 2.523229693228642941  --> 12327.69765106043423
14 -> 2.570522369163364754  --> 31688.62257233351406
15 -> 2.614653358199854440  --> 82854.76342547953234
16 -> 2.656008810659357128  --> 220062.9816631702952
17 -> 2.694903007137808401  --> 593048.3910437900173
18 -> 2.731581618923209325  --> 1619960.084107200449
19 -> 2.766272350855856453  --> 4481250.790155876330
20 -> 2.798806174651727408  --> 12542152.38165119906
21 -> 2.797113371058406802  --> 35081822.12856861096
22 -> 2.423148937773094341  --> 85008480.02598566512
23 -> 1.541937578649181143  --> 131077769.8559156157
24 -> 0.003825393288703888199  --> 501424.0211050924189
25 -> 5.443317760119889255 E32  --> 2.729410279432079743 E38