A Recursive Function Problem in Mathematica

Hello,

I have the following recursive function:

where

and I tried to write it using Mathematica as following:

Code:

**fun[m_,n_,r_]:=If [m<=r<=n,1,0];**

**B[0,j_,N_]:=1;**

**B[1,j_,N_]:=j;**

**B[k_,1,N_]:=1/k!;**

**B[k_,j_,N_]:=Sum[B[l,j-1,N]/Factorial[k-l]×fun[0,((j-1)×(N-1)),l],{l,l=k-N+1,k}]**

I have this function programmed well in MATLAB, and I use it as a reference to check the Mathematica code which fails to give correct answers. Can anyone tell me where is the error in the Mathematica code?

**Hint**: If you write at Mathematica B[3,3,3] it must give you 4, B[2,3,3] = 4.5, B[4,4,4]=10.5 as given by using the MATLAB code.

Thanks in advance