This gun would have been imported c. 1890-1915 by H&D Folsom of New York. You won't be able to narrow it down any closer. The number 1610 is probably a manufacturer's batch, lot, catalog, or part number. If the gauge is marked, it will be expressed in centimeters, like 18.0 65. That example would mean a 12 gauge, full-choke, with 2 9/16 chambers.
From the late 1800's until WWI thousands of shotguns like this were imported from Belgium. If the country of origin is marked on the gun, it was probably made after 1893 when this was required by international trade agreements. The companies that retailed or distributed some of the trade names have been identified, but I don't find Hanover on that list. Although these guns were safe enough when new, the damsacus barrels may have deterioriated and should be retired. They usually bring $125-$150 as mantle decorations.
Diamond Arms Company: Trade name used by the Shapleigh Hardware Company of St. Louis, Missouri on imported shotguns they retailed. WWI pretty much ended these imports and later guns may have been made by any of several US companies. A single shot is possibly an Iver Johnson Champion, a double barrel likely to be by Crescent or Stevens. There is little collector interest in these old utility guns and shooters would rather have newer firearms, so they are not highly valued. Singles sell for $50-$75, doubles for $125-$150.
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