My researches have led me into the area of radio controlled models. They have much the same problems as electric bikes; a need for light, high-energy batteries at a good price.
The first thing I had to learn about was C. C is coloumbs, a unit I learned about 50 years ago and now have no memory of. It's a unit of electrostatic charge, but it also expresses how fast you can get energy out of a battery. Suppose you have a battery that is 2 amp-hour (which is 7200 coloumbs). then if that battery is 20C (a common value for Li-ion) then it can deliver 20 amps continuously. I hope I have that right. So a 20C, 10AH battery can deliver 200 amps. I need about 20 amps at most.
So I'm looking at the Turnigy 5000mAh 4S1P 14.8v 20C. That gives me 5 amp-hours at 14.8 volts, and it can give 100 amps. If I put two of those in series, and another pair like it in parallel, that's 10 amp-hours at 29.6 volts with up to 200 amps at a cost of £68 (the last time I bought a 24v, 10ah battery, it cost £196). And the weight is 528 grams, so four is 2.16 kg (my existing batteries are 3.12 kg).
Now for charging. It's not just a matter of bunging 30 volts to it and waiting for it to charge - you have to be really careful with Li-ion.
I want to be able to charge it overnight, say 10 hours. that means charging each of the four units at 1/2 amp, to give the 5 ah. That's 2 amps at 14.8 v, which is 30 watts. So I need a 30 watt charger. But I'd like to double that so that I can use twice as big a battery, so the charger I want looks like the HobbyKing ECO8 150W 7A 8S costing £30; that will give plenty of watts and amps, and lets me charge 8 cells at a time, while simultaneously balancing them. And to power that, I'll use a PC power supply (cost zero because I have loads of them). This is my Battery Management System; used on charging, which is when the main issues happen, but not on discharging, which means it doesn't need to be a permanent part of the battery.
So, balancing. If you just put 15 volts across the 4-cell 14.8 volt Turnigy, the four cells that make it up, won't charge evenly, and that could be bad, even catastrophic. So what this charger does, is charge each cell separately so that they all get taken up to 4.2 volts. I think I can do all four of the Turnigy batteries in parallel at the same time, if I use a splitter costing a couple of pounds. I'd package the splitter up with the batteries so I can just plug the charger into that, put in a car fuse on the output that will blow at 25 amps (for belt-and-braces safety) and take the output to a kettle plug that will plug into my bike motor controller. I won't need a switch, because unplugging the kettle has the effect of a switch.
The cells are nominally 3.7v, hence the 29.6v total. But when fully charged they are 4.2v, for a total of 33.6v. I hope the bike can handle that! If I let them go down to 3.0v then that's 24v total, so that would be a good point to cut off the power and put in a new battery.
What I don't know, is whether that 5000 mAh is real. Because when I was looking at the 18650 cells, the manufacturer's boast was usually somewhat more than reality; sometimes a lot more. More research required.
Also, are the batteries protected against over-discharge? If you over-discharge a Li-ion, that leads to it never working again. More research required. But £2.66 gets me a loud low-voltage detector which will monitor each cell of on battery, so maybe I get four of those.Plus I have my voltmeter on the handlebars so I can monitor as I ride.
Hobbyking also sells a "Lipo bunker", a steel case that you can put the battery into while charging if you're paranoid about the possibility of it overheating. It looks a bit like an ammo can ... and I can use one of my ammo cans for this!
So, this is looking to be a better proposition that 50-odd 18650 cells.