Yes, Remington did make a little-known and seldom seen rifle in 44-40 (also 38-40). It was a pump rifle called the Model 14-1/2. It was very much like the more often seen Model 14 and Model 141. This model was made from 1912 to 1925. It was made in rifle (22-1/2" barrel) and carbine (18-1/2" barrel). The model never caught on with the public for many possible reasons. It was directly competing with the very popular and well-made Winchester model 1892 lever action (in both rifle and carbine formats), and also the Marlin Model 1894 lever action (in both rifle and carbine formats). The lever action had a well-established following but the pump action guns were just getting off the ground. The 44-40 and 38-40 cartridges were well-known and popular but were black powder based and losing popularity by this time.
People wanted the newer more powerful smokeless powder designed cartridges. While these Remington guns were accurate and well-made, their fate was sealed. The rifle seen in these pictures is the rifle version (22-1/2" barrel) in caliber 44-40 Winchester. Of course Remington did not want to promote the Winchester name, so they said the caliber was 44 Remington (even though there was no such cartridge). Remington did do an interesting thing with all of their center-fire pump rifles of this time period. They inset a brass cartridge head into the left side of the action. This was a decoration and a handy caliber reference for the shooter.
These guns had a somewhat unique location for loading the ammo. It has a loading gate on the bottom, located between the receiver and the forearm. This barrel length has a magazine capacity of 11 shells. The action release button is located in the not-too-handy location of the back right side of the bolt. You push this button to open the action. The safety is the common Remington button behind the trigger. These guns had a simple take down feature of the single, large screw on the left side of the action. This broke the gun down into two pieces of roughly the same length for cleaning or transport.
The pictured gun has a vintage Lyman Tang sight added (these guns came tapped for these sights). I put the original rear sight back on the gun for these pictures (you can only use one or the other of these rear sights at a time). The peep sight is better for us older folks with tired eyes. On this rifle, stamped on the left side of the barrel near the action, is the letter "M". This is Remington's date code for a manufacture date of 1921.
This rifle, while far from prime condition, does sport an excellent bore and is a very good shooter. Yes, I have spent many shooting sessions at the range with it. The 44-40 cartridge is a joy to shoot with its low noise and very low recoil. While this cartridge has taken many deer and other game in its long (way over one hundred year) life, it is about the minimum power for even very close range deer. This is just one of many old guns that history has all but forgotten.