newptsxy[PTS_] := Block[{ut, i, n},
    (n = Length[PTS];
      If[n == 1, ut = PTS, 
        ut = Table[y PTS[[i]] + x PTS[[i + 1]], {i, 1.n - 1}]];
      Return[ut])]


mkpctxy[PTS_] := Block[{i, SEQ, te, n, prev}, (n = Length[PTS];
      SEQ = {PTS};
      prev = PTS;
      Do[(te = newptsxy[prev];
          SEQ = Append[SEQ, te];
          prev = te;), {i, 1, n - 1}];
      Return[Map[Expand, SEQ]])]

mkpctxy[{P0, P1, P2, P3, P4}]


newpts[PTS_, t_] := Block[{ut, i, n},
		   (n = Length[PTS];
		     If[n == 1, 
					          ut = PTS,
					          ut = Table[(1 - t) PTS[[i]] + t PTS[[i + 1]], {i, 1.n - 1}]
					       ];
				 Return[ut])]

mkpct[PTS_, t_] := Block[{i, SEQ, te, n, prev},
		 (n = Length[PTS];
		 SEQ = {PTS};
		 prev = PTS;
		Do[
			  (te = newpts[prev, t];
					SEQ = Append[SEQ, te];
				  prev = te;
				 ), {i, 1, n - 1}];
		Return[SEQ])]

FF = mkpct[{{1, 2}, {5, 13}, {9, 14}, {16, 12}, {22, 1}}, t]

Map[Print, %]

boo = Map[Expand, FF[[5]]]

ParametricPlot[Evaluate[boo], {t, 0, 1}]


mkdisks[PTS_, r_, c_] := 
  Table[{Hue[c], Disk[PTS[[i]], r]}, {i, 1.Length[PTS]}]

disp[PTS_, t_, c_] := Block[{i, m, CPS, ut, Ls, Ds, LAP},
		     (CPS = mkpct[PTS, t];
			   m = Length[CPS];
			 LAP = CPS[[m]][[1]];
			  ut = {};
			  Ls = 
        Table[{Hue[c/(i^.2)], Thickness[.007], Line[CPS[[i]]]}, {i, 1, 
            m - 1}];
			 Ds = Table[mkdisks[CPS[[i]], .3/(i^.3), c/(i^.3)], {i, 1, m}];
			 Show[Graphics[Join[Ls, Ds, {Disk[LAP, .3]}]], AspectRatio -> Automatic,
        Axes -> True, ImageSize -> 71 6]) ]


disp[{{1, 5}, {3, 13}, {7, 16}, {13, 15}, {17, 8}}, .4, .9]


<< Graphics`Animation`

seqraph[PTS_, t_, c_] := Block[{i, m, CPS, ut, Ls, Ds, LAP},
		     (CPS = mkpct[PTS, t];
			   m = Length[CPS];
			 LAP = CPS[[m]][[1]];
			  ut = {};
			  Ls = 
        Table[{Hue[c/(i^.2)], Thickness[.007], Line[CPS[[i]]]}, {i, 1, 
            m - 1}];
			 Ds = Table[mkdisks[CPS[[i]], .3/(i^.3), c/(i^.3)], {i, 1, m}];
			 Graphics[Join[Ls, Ds, {Disk[LAP, .3]}]]) ]


ShowAnimation[
	 Table[
		   seqraph[{{1, 2}, {5, 13}, {9, 14}, {16, 12}, {5, 5}, {10, 1}, {15, 4}}, 
      i/50, .9], {i, 1, 49}]
]