Playing around with this a bit and trying different values of k suggests that the general formula for the i-th moment is given by
m[i_, n_, k_, \[Theta]_] := - FactorialPower[n, i]^k \[Theta]^i HypergeometricPFQ[ConstantArray[i - n, k],
ConstantArray[i + 1, k - 1], (-1)^(k - 1) \[Theta]/(-1 + \[Theta])]/
((i!)^(k - 1) (-1 + \[Theta])^i (-1 + HypergeometricPFQ[ConstantArray[-n, k], ConstantArray[1, k - 1],
(-1)^(k - 1) \[Theta]/(-1 + \[Theta])]))
