What you have there is more commonly known as the MODEL 755 SAHARA. It is a single shot "semi-automatic" youth rifle. This is a simple "blow back" design that automatically ejects the spent casing and locks the bolt back for reloading. I can't find a reference to when H&R called this the "HUNTSMAN" - but the stamping on the barrel irrefutably labels your gun as that. Another mystery semi-solved. We know H&R made these and they were offered, in the catalogues as the "SAHARA Model 755" from 1963 to 1970. I have one in my collection (see pics) and have seen a very limited number of others over the years (less than a half dozen) so I'm guessing they were not made in large quantities.
The lack of a serial number reflects the "low cost" of these rifles and the fact that s/n were not required to be placed on guns before the GCA of 1968. So your piece was made before 1968, and as early as 1962-63. My piece has an "AF" prefix to a serial number, indicating manufacture in 1969. I'm guessing again, but it would be logical to assume that the "HUNTSMAN" name may have been used very early in production and then changed at some point - perhaps when they were first catalogued????????