nmax := 10;
pascal[nmax_] := Module[
   {vals = Table[Binomial[n, k], {n, 0, nmax}, {k, 0, n}],
    ids = Table[{n, k}, {n, 0, nmax}, {k, 0, n}],
    labels, left, right, leftright, edgeLabels
   },
   labels = Flatten[Thread /@ (Thread[ids -> vals]), 1];
   left = DeleteCases[Flatten[Flatten[ids, 1] /. {n_, k_} /; (n >= k + 1) :> {{n, k + 1} -> {n + 1, k + 1}}, 1], _?NumberQ];
   right = DeleteCases[Flatten[Flatten[ids, 1] /. {n_, k_} /; (n > k) :> {{n, k} -> {n + 1, k + 1}}, 1], _?NumberQ];
   leftright = DeleteCases[left \[Union] right, _ -> {b_, _} /; b > nmax];
   edgeLabels = (# -> Style["+", Medium] & /@ leftright);
   Graph[Flatten[ids, 1], leftright
    , VertexLabels -> MapAt[Placed[#, Center] &, labels, {All, 2}]
    , GraphLayout -> "SpringEmbedding"
    , VertexSize -> 0.8, EdgeLabels -> edgeLabels
    , PlotLabel -> "Pascal's Triangle"
   ]
  ];
pascal[nmax]