I wonder whether this one might be equivalent to a known infinite product for pi or pi/4 or something. The ratios between successive convergents are all of the

I wonder whether this one might be equivalent to a known infinite product for pi or pi/4 or something. The ratios between successive convergents are all of the form (a_n+k_n)/a_n where k_n is 1, 1, 3, 3, 15, 15, 105, 105, 315, 315, ... -- i.e., the least common multiples of odd numbers up to n. [EDITED to add:] This should make you think of the Wallis product formula for pi/4, though this doesn't seem to be the Wallis product in disguise. Clearly these k_n divide n!, so maybe the right way to say this is that the conjecture seems to be equivalent to 4/pi = product (a_n+n!)/a_n where (a_n) = (4,130,2464,45448,882528,18410640,...) ... though I don't know what that sequence _is_, haven't shown that it has a nice form, etc. None of (a_n), (a_n+n!), (a_n+n!/2) seems to occur in OEIS or to be a subsequence of anything in OEIS.