negative binomial coefficents

Clear[nbinomial, nbinomial2]
(*$Assumptions = m ∈ Integers  ;*)

nbinomial[b_, k_] = (-1)^k Binomial[-b + k, -b] (-b)/(k-b)
(*nbinomial2[b_, k_] = (-1)^k Gamma[-b + k]/(Gamma[-b ] Gamma[k+1]) /. Gamma[p_] -> (p-1)!*)

(*Sum[nbinomial[-2, k] a^(-2 -k) x^k, {k, 0, 10}]

Sum[nbinomial2[-2, k] a^(-2 -k) x^k, {k, 0, 10}]

Series[(a + x)^(-2), {x, 0, 10}]*)