Oleg has a very interesting approach; in particular, he avoids explicit recursion and (most) computing with rationals, while also replacing the factorials in binary coefficients by using successive rows of Pascal's triangle. He also skips over the B_{2k+1}, k > 0, which are all 0. I slogged through the "standard" expansions, deriving a tail recursive function that rolls along successive Bernoulli numbers, generating successive rows of Pascal's triangle along the way, and returning the list of B_n .. B_0. You can think of the list of Bernoulli numbers as "memoizing" just-in-time. Running it in Hugs on a 650Mhz Athlon with 128M RAM, bernoulli 82 took ca. 1 sec. Compiling with ghc -O, bernoulli 1000 took approx. 15 sec. wall time; bernoulli 10000 blew the heap. I couldn't get getCPUTime (from module CPUTime) to work for me; if anyone can enlighten me on how to get timing of function execution I'd appreciate it. BTW profiling didn't work; when I tried to compile with profiling flags I received error msgs saying that interface files for standard libraries couldn't be found. Compiling without the flags seems to work just fine. Oleg's program brings up another interesting point for all you mathematicians out there: I found two basically different expansion formulas for Bernoulli numbers. One is based on the formal expansion of the equation (B + 1)^n = B^n which results in binomial-theorem expansions all of whose terms are positive. The other is based on formal expansion of the equation (B - 1)^n = B^n which results in binomial-theorem expansions whose terms alternate in sign. The amazing thing is, they two sets of numbers only differ at one term: the first formula results in B_1 = -1/2 while the second results in B_1 = +1/2 ! I found the second formula in Conway & Guy, _The Book of Numbers_, p.108. The second formula, with tiny variations, can be found in Knuth, _Fundamental Algorithms_, p. 109, Abramowitz & Stegun, _Handbook of Mathematical Functions_, p. 804 and Song Y. Yan, _Number Theory for Computing_, p. 75 This has gotten a little long, sorry. If you want I can post my Haskell code or send privately. -- Bill Wood wtwjek@winternet.com