I used to have flats... the last one was April 2004 and before that, September 1999. That looooooong stretch of time consisted of using kelvar BELTED tires (not kevlar beads) thorn resistant tubes for a year of that (in Korea) and I don't use racing tires.
Presently, I use a 20x1.50 (406x40) Schwalbe Marathon 100 PSI and 700Cx35 (622x37) Schwalbe Marathon Slick 87 PSI on my recumbent. My wife has been running Marathons on her recumbent for 1.5 years and has never had a flat.
While riding across Oklahoma on a week-long group ride, I heard a loud pop and crunching sound. A goat-head thorn punched into the tire, hit the belts and was crushed...no flat!
If you really don't want any flats, Schwalbe makes this monster of a tire called the Marathon Extra or something along those lines. It has a very, very thick soft stuff under the treads and it is pictured with tacks stuck in the tire. /ubbthreads/images/graemlins/faint.gif They are heavy but it takes a nail gun to get flats from what I see.
I had a friend that put "air-free" things on his recumbent (he weighed 275 at the time) The problem with those things besides weight, bone-rattling ride and poor traction was they trashed his rims! When those things get hit by a pot hole, they transfer the force directly to the rim instead of cushioning the blow. After he replaced his rear wheel, he went to kevlar tires and flats rarely.
My battle wagon setup for Korea was rather insane. I used a Specialized Nimbus Armadillo (kevlar belts)35x700C and thorn resistant thick inner tubes for the rear. On the front, a Snafu Rim Job (that is the name...gotta love Freestyle names) that tire was 130 PSI 20x1.95 with kevlar belts. That combo handled riding in over 500 meters of glass. I was wondering why the dirt road had a greenish sheen and rounded a corner--straight to mountains of crushed glass at a recycling center! I picked up my recumbent, brushed the tires off and carried it away. No flats ever with that combination.
I had one flat in the last 5.5 years, a construction tack punched through my back tire so what are you going to do? My first year of riding my recumbent, my rear 700Cx28 kevlar belted tire flatted 3 times. My front Schwalbe City Marathon 20x1.25 115 PSI tire had kevlar belts also, ran it for 1.5 years without a flat.
To make this loooooooong post clear, run slightly wider kevlar belted tires and if need be, thorn-proof inner tubes. I have had great luck with Specialized Armadillo and Schwalbe Marathon series of tires. The 700Cx28 was an IRC so I avoid those.