Falcon Heavy is expected to take 4 times more mass to LEO than Falcon 9 (53 instead of 13 tons). Are bigger rockets generally about 4/3 times more capable per fuel mass and engine thrust? It doesn't seem to be true for Delta IV Heavy compared with Delta IV. Its 3 common core boosters give 3 times more payload mass to LEO. (9 and 27 tons).
The 53 metric tonne number comes from the use of propellant cross feed. And only when using propellant cross feed. When not using it, payload to LEO is lower, but I cannot find a reference to that number.
Cross feed means that the center core has connections to the side cores, sending fuel and oxidizer over from each side to the engines of the center core. This way, when the two side boosters burn out, the center core is mostly full of fuel/oxidizer, converting from a 2 stage vehicle to a functional three stage vehicle. The side boosters only feed the three engines closest to the side booster. Thus 6 of the 9 engines are cross fed, and the middle three of the middle core are fed from its native tankage.
Then the center core, acting as a second stage, can burn most of a full Falcon core's worth to boost the upper stage (now really a third stage) to orbit for it to finish the mission.
It is amazing how big a deal cross feed makes. Alas, it turns out that SpaceX decided it was not worth the effort. Instead they went with launching with all 27 engines at full thrust, and soon after they are moving, throttle down the 9 engines from the middle core, so that the side cores burn out sooner and still give a staging effect.
Delta 4 Heavy, burns all three cores at basically the same rate, though they might be considering full thrust to start, throttle down the center core, separate the cores, and burn the center core a bit longer to get a pseudo third stage effect.
Angara 3 or 5 which will be flying with multiple strapons may have the ability to do something similar with throttling, but unless it is built in from the beginning, cross feed is very hard to add afterwards.
But throttling down the center core to burn longer after the side cores separate is not as effective of emptying the side boosters much faster, staging earlier, throwing away all that extra mass, and then burning the lighter 'second' stage longer. However, SpaceX decided to go with throttling instead of cross feed in the end.
Most likely this is because no paying customer actually needed the full payload capacity for a mission. Thus they get close enough with throttling. Additionally, between the time they announced the first iteration of Falcon Heavy, and the first launch, the Falcon core booster greatly improved.
The performance of the Falcon 9 1.0 (the first 5 boosters) differs greatly from the Block 5 Falcon 9. They went from a LEO payload of 10,450kg to 22,800kg and the penultimate Falcon Heavy will be using the newer boosters.
The first Falcon Heavy mission actually uses Block 3 side boosters (They are previously flown Block 3 boosters). The center core is unique and new, but is thought (Not seen any evidence yet) to not be quite the Block 5 design. The future flights are expected to use three Block 5 cores.
Much of that performance boost is Merlin related, where the Merlin 1A/1B from the original Falcon 1 was only around 75,000 lbs of thrust.
The Merlin 1C used in the Falcon 9 1.0 was a great improvement at closer to 125,000lbs of sea level thrust.
The Merlin 1D used in Falcon 9 1.2/Full Thrust is closer to 170,000lbs of thrust.
The final version expected to be used in Block 5 (No name change officially yet) is expected to increase again, closer to 190,000lbs.
Thus they more than doubled single stick payload performance between the time Falcon Heavy was announced and its first launch.
This allowed them to get payload back to what they originally predicted, without the need for the more complex cross feed.
https://en.wikipedia.org/wiki/Falcon_9 Falcon 9 payload to LEO: 22.8tons to GTO: 8.3tons
https://en.wikipedia.org/wiki/Falcon_Heavy Falcon Heavy payload to LEO: 63.8tons to GTO: 26.7tons
FH performance is just 180% more for LEO, but 220% more for GTO
So, no, its NOT 4x more to LEO, that was using cross feed, which has been abandoned because its too risky and complex and is not necessary for existing missions SpaceX sees enough market opportunities for
With cross feed, FH might throw 70+ tons to LEO, perhaps over 4x as much
Why is that ? The two additional side boosters don't add performance linearly. The 3 boosters function as FH first stage and then when the side boosters are jettisoned, the center booster alone functions as a second stage. This improves performance non linearly. Without cross feed, the side boosters must stay attached longer, reducing the burn time of the 2nd stage configuration. With cross feed, the side boosters feed the center booster fuel until separation, providing a fully fuelled center booster upon separation, substantially increasing the 2nd stage burn time In this thought process the upper stage is now a 3rd stage
The further reason cross feed was abandoned is with it the center booster finishes burning going way too fast and way too high, that in order to recover it, an ASDS platform might have be be positioned 2x to 3x as far from the launch pad as with F9. Since the revised Block V non cross feed FH can now handle all missions SpaceX has been hired to perform, its logical to make it easier to recover all 3 boosters which is far more important than having performance no customer is willing to pay for.