Sage is using currently Maxima behind the scenes for calculus-type problems at the moment.  At the moment, Gary Furnish (an undergrad at University of Utah) is working on some more native calculus support in Sage with the intention of using it as a base for doing differential geometry computations.

--Mike

The code to compute the number of partitions was written by Jonathan Bober who is a grad student at Michigan.  It is fairly highly optimized C code that uses the GMP, MPFR, and quaddouble libraries.

--Mike

P.S.  Using 5 lines in Sage, I was able to get the following additional terms for the sequence:

30270, 31679, 31720, 32890, 33081, 33554, 34027, 35090, 35231, 35833, 37317, 38058, 38868, 38950, 38958, 39183