It is not clear to me what the relation is between n and j, and whether k1,k2 etc can be zero. However you can get started by looking at how this is built:
g[j_, d_, n_] :=
Plus @@ Table[
Times @@ Map[Gamma[d + 2 #]/(#! Gamma[d/2]) &, k], {k,
IntegerPartitions[j, {n}]}]
Look up the key ingredient, which is IntegerPartitions