try slanting the top of the bridge towards the tailpiece just a bit. that did it for me

It sounds as if the nut slot is cut too wide or deep and is either vibrating in the slot or against a fret wire. Look at the stings in the nut and see of one of them is sitting lower than than the others. Slot can be filled with baking soda and super glue and recut. You have to know what you are doing though.
Alternatively you can take it back to where you purchased it (if possible) and it should be repaired under warranty.

Hard to say exactly what you do next, because there's still more than one thing it could be.

The nut slot is certainly the leading suspect. Maybe too deep or too wide.

If too deep, then maybe the string was buzzing one or more of the low frets.

If the issue is the nut, then maybe the banjo goes back to the store so their repair people can deal with it. On a new banjo, a new nut should be offered. Slots can also be filled with baking soda and super glue and refiled, but I don't think that's appropriate for a new instrument under warranty.

Was the bad sound only on the open first string and disappeared at any fretted note? Or was it also not good in some of the low frets but fine above the fifth?

If the second was the case, it could be the neck is too flat and the truss rod needs a very small (1/6 to 1/2)  turn to introduce some "relief" (up-bow) in the lower frets. You measure relief by capoing the strings at the first fret, depressing a string at the 17th fret or higher, then using a feeler gauge to measure the clearance between a string and the 7th fret. Measure the 1st, since that's the problem string. If you don't have feeler gauges, then a business card will do for determining if the clearance is at least .01-inch.

There's no one right amount of relief for all banjos. It's whatever makes the buzz in low frets go away without strings being too high in the middle frets. I think most banjos do OK with relief of .012 to .015.  But some need as much as .02.  Stelling specs are for even more.

Good luck.

Nut slot, nut angle, bridge placement, /slot probably the problem. Capo it at first fret and see if it still makes that Sound . If so ,it could be the bridge ? If the sound goes away its probably in the nut . Best of luck

New banjo. Was ok, now it’s not. I’m betting the new banjo was set up with the head tightened to 90 - 91 on the drum dial, or approx g#. New heads loosen up, and need tightened several times in the first few months, before settling down. After a few weeks/month or so, when the head stretches, and gets close to 88-89 on the drum dial, it gets close to a g note, and sympathetic vibrations start making strings sound weird, and sitar like. If you tighten each j hook nut 1/2 turn, I bet the weirdness goes away.

Could be fret end lifting due to weather changes

Head tension first.
Neck bow/truss rod.
I agree with .012-.015 and Stellings being in the .20s.
A neck can be low on the 1st string side and need to be "leveled" or lined up with the hoop.
The likeliest are relief and head tension.

Might want to check string diameters.

The banjo was ok. 

Then the first string started sounding bad.

Then putting a piece of paper in the bridge slot under the first string solved the problem. 

Other strings sound ok. 
 

What does this evidence point to?

What does the evidence eliminate?

Has a new first string been tried?

Something else to check, that I have seen do this. Make sure the neck is tight to the pot. Especially on new banjos, where the wood is still settling, and who knows how well the parts were tightened in the factory. Simple, lay the banjo down and grab the neck and twist it, and see if it moves. If the neck has twisted on the pot recently, it could lower the action on the 1st string side, until the 1st string starts buzzing. Adding the paper might have raised it enough to stop it. Definitely something to check and eliminate.

Adding something under a string to fix the sound might just be an indication of a deeper problem.

Mt guess is, as it has been suggested, that the bridge slot for the first string is either too wide or too deep, since a piece of paper helps. How low is the action of the first string? Maybe another bridge will help, and maybe (but not necessarily) a little higher. Otherwise - keep the paper. Hope you have already found a solution. Steen

Just a long shot, but look at the angle of your right hand . Sometimes the middle finger winds up hitting the first string way closer to the bridge than the others do. Producing a twangy sound. Check to see if that middle finger pick is hitting the first string nice and flat and not towards side edge of the pick. Just something to check

Put on a 1st string that is bigger than the one on it.
If it's a .009 go to a .010.
If it's a 10 go to an 11.

Oooops! ... edited out an ill-advised post.  And I offer up a Canadian "Sorry."