I'm not sure what type they are, but it is similar to the MacArthur palm (Ptychosperma macarthurii) - the similarties are that it grows in clusters, and has a similar seed thing, but I never remember seeing red seeds on it. I'd say it's definitely some kind of Ptychosperma, but which one I'm not sure...
A few days ago, I noticed that the seed/flower stalks had burst into bloom - I'd never noticed the flowers there before. Maybe it's the recent hot and dry weather.
Closer up, I saw that it was buzzing with bees!
On this one you can see the pollen that has been collected (a 'pollen sac'?)
Based on the picture of the flowers
In fact, I have taken to pulling off the flower stalk, as my father-in-law suggested - in order to stop lots of little trees growing up. But now I feel kind of guilty, having seen the bees enjoying the flowers so much!
It reminded me of one of the paradoxes of gardening - I like to think of myself as using the garden to get closer to nature, using organic means, caring for the environment and all that; but in the end of the day, a garden is such an unnatural environment :/