(didn't need the l= syntax):

Code:In[1397]:= 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, k - n + 1, k}] B[3, 3, 3] B[2, 3, 3] B[4, 4, 4] Out[1402]= 4 Out[1403]= 9/2 Out[1404]= 21/2