These interesting coins were struck by Henri himself, not by France. Henri came close to being king, and he had credentials, but never quite made it. He was a pretender. Henri of Artois (1820 - 1883) was disputedly King of France from 2 to 9 August 1830 as Henry V, although he was never officially proclaimed as such. He was also pretender to throne from 1844 to 1883. He was the son of Charles Ferdinand, a son of Charles X of France.
Henri issued coins in 1 and 5 franc denominations. The 5 francs are quite a bit more valuable:
worn: $50 US dollars approximate catalog value
average circulated: $80
well preserved: $120
fully uncirculated: $180
worn: $250 US dollars approximate catalog value
average circulated: $800
well preserved: $1200
fully uncirculated: $1800
Requester Adam provided a great picture of his coin and we have used it on this page. Adam's coin is in well preserved condition with good eye appeal, but there are tiny scratches on Henri's portrait. Our guess is that Adam's coin would sell around $100 to an interested collector. A coin dealer would probably offer $50 for it.