mlb>Wow! These are like Michael Somos's unreasonably-integral sequences
on steroids.
Shallower, I fear. The recurrence I gave for F coincides with a three- term recurrence for a terminating 2F1 with entirely integral terms. And, to my mild embarrassment, the 3F2[-4] = 1,6,57,701,10147,164317,2888282, can be written n ==== \ n (n + 2 i + 3)! > 2 ( ) --------------------------, / i (i + 1)! (i + 2)! (n + 3)! ==== i = 0 which likewise appears to be termwise integral. (It would be obvious (and even) if that numerator were (n+2i+6)!, e.g. We'd have a nice, Pascaloid recurrence.) So why is this summand an integer. Or is it? --rwg