Pochhammer[(1/2), k]

Needs["SpecialFunctions`"]

{HolonomicRE[Pochhammer[(1/2), k], a[k]], 
      Pochhammer[(1/2), k] /. k -> 0}

{HolonomicRE[(((2  k)!)/((4^k) k!)), 
        a[k]], (((2  k)!)/((4^k) k!)) /. 
        k -> 0}

Pochhammer[(1/2), k]

RE = HolonomicRE[Binomial[n, k], a[k]]

(Subscript[RE, 1]) = 
      HolonomicRE[Binomial[n - 1, k] + Binomial[n - 1, k - 1], 
        a[k]]

(Subscript[RE, 1]) /. a[k_] -> Binomial[n, k]

(Subscript[RE, 1]) /. a[k_] -> Binomial[n, k] // FullSimplify

{Binomial[n, k], Binomial[n - 1, k] + Binomial[n - 1, k - 1]} /. 
      k -> 0

{Binomial[n, k], Binomial[n - 1, k] + Binomial[n - 1, k - 1]} /. 
      k -> 1

(Subscript[RE, 2]) = 
      HolonomicRE[Binomial[n - 1, k] + Binomial[n - 1, k - 1], 
        a[n]]

(Subscript[RE, 2]) /. a[n_] -> Binomial[n, k] // FullSimplify

{Binomial[n, k], 
        Binomial[n - 1, k] + Binomial[n - 1, k - 1]} /. {k -> 0, 
        n -> 0}

{Binomial[n, k], 
        Binomial[n - 1, k] + Binomial[n - 1, k - 1]} /. {k -> 0, 
        n -> 0}

{Binomial[n, k], 
        Binomial[n - 1, k] + Binomial[n - 1, k - 1]} /. {k -> 1, 
        n -> 0}

{Binomial[n, k], 
        Binomial[n - 1, k] + Binomial[n - 1, k - 1]} /. {k -> 1, 
        n -> 1}