I'll guess that some here would dispute that this series qualifies as "converges rapidly", but it seems worth passing along the challenge (to prove the conjecture.) Rich -----Original Message----- From: Number Theory List [mailto:NMBRTHRY@LISTSERV.NODAK.EDU] On Behalf Of Zhi-Wei SUN Sent: Thursday, January 13, 2011 8:35 AM To: NMBRTHRY@LISTSERV.NODAK.EDU Subject: A conjectural series for 1/pi of a new type Dear number theorists, On Jan. 2, 2011 I found a new type series for 1/pi. CONJECTURE (Zhi-Wei Sun). We have Sum_{k=0,1,2,...}(30k+7)binom(2k,k)^2*a_k/(-256)^k = 24/pi, where a_k=T_k(1,16) is the coefficient of x^k in (x^2+x+16)^k. I have included this conjecture in an article of mine (to be expanded later) available from http://arxiv.org/abs/1101.0600 Since a_k=T_k(1,16) ~ 0.75*9^k/sqrt(k*pi) as k tends to the infinity, we have binom(2k,k)^2*a_k/(-256)^k ~ 0.75(-9/16)^k/(k*pi)^{1.5} and hence the series in the conjecture converges rapidly. I have contacted several famous experts at pi-series or modular forms, they have never seen such a series for 1/pi before and none of them could prove the conjecture. It seems that all known methods used to prove Ramanujan-type series for 1/pi (including the current theory of modular functions and the WZ method) do not work for this curious series. Such a new series for 1/pi should be very rare! I consider the conjecture particularly difficult and very challenging. It might appeal for a powerful tool or a new theory. Any comments are welcome! Zhi-Wei Sun http://math.nju.edu.cn/~zwsun