Double down is a weapon perk that acts similar to the innate sidearm perk. The sidearm perk gives you ammo every time you die in destiny. The double down perk is suppose to "occasionally" give ammo but I have noticed it never works in the crucible for me. I haven't tried any weapons with this perk in any sort of pve activities.
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Double Down only works off of the initial spawn in a Crucible Match. Not after a death.
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It only works when you first spawn in at the very beginning. This is the same for all weapons that has double down.