Generalization of rule 30

(*triangle formatting*)
(*start*)
Clear[t, n, k]; t[n_, 1] = 0; t[n_, 2] = 0;
t[n_, 3] = 1;
t[n_, k_] :=
 t[n, k] =
  If[2*n + 1 >=
    k, ((1 - (t[n - 1, k - 2]*t[n - 1, k - 1]*t[n - 1, k]))*(1 -
        t[n - 1, k - 2]*
         t[n - 1,
          k - 1]*(1 - t[n - 1, k]))*(1 - (t[n - 1,
           k - 2]*(1 - t[n - 1, k - 1])*
          t[n - 1,
           k]))*(1 - ((1 - t[n - 1, k - 2])*(1 - t[n - 1, k - 1])*(1 -
             t[n - 1, k])))) + ((t[n - 1,
         k - 2]*(1 - t[n - 1, k - 1])*(1 - t[n - 1, k]))*((1 -
          t[n - 1, k - 2])*t[n - 1, k - 1]*
        t[n - 1, k])*((1 - t[n - 1, k - 2])*
        t[n - 1,
         k - 1]*(1 - t[n - 1, k]))*((1 - t[n - 1, k - 2])*(1 -
          t[n - 1, k - 1])*t[n - 1, k])), 0]; TableForm[
 ArrayPlot[Table[Table[t[n, k], {k, 3, 2*n + 1}], {n, 1,
   100}]]] (*Mats Granvik, Dec 06 2019*)
(*end*)

(*pyramid formatting*)
(*start*)Clear[t, n, k];
nn = 100;
t[1, k_] := t[1, k] = If[k == nn, 1, 0];
t[n_, k_] :=
  t[n, k] = ((1 - (t[n - 1, k - 1]*t[n - 1, k - 0]*
          t[n - 1, k + 1]))*(1 -
        t[n - 1, k - 1]*
         t[n - 1,
          k - 0]*(1 - t[n - 1, k + 1]))*(1 - (t[n - 1,
           k - 1]*(1 - t[n - 1, k - 0])*
          t[n - 1,
           k + 1]))*(1 - ((1 - t[n - 1, k - 1])*(1 -
            t[n - 1, k - 0])*(1 - t[n - 1, k + 1])))) + ((t[n - 1,
         k - 1]*(1 - t[n - 1, k - 0])*(1 - t[n - 1, k + 1]))*((1 -
          t[n - 1, k - 1])*t[n - 1, k - 0]*
        t[n - 1, k + 1])*((1 - t[n - 1, k - 1])*
        t[n - 1,
         k - 0]*(1 - t[n - 1, k + 1]))*((1 - t[n - 1, k - 1])*(1 -
          t[n - 1, k - 0])*t[n - 1, k + 1]));
ArrayPlot[Table[Table[t[n, k], {k, 1, 2*nn}], {n, 1, nn}]]
(*Mats Granvik,Dec 07 2019*)
(*end*)

(*entries as polynomials*)
(*start*)
Clear[t, n, k];
nn = 4;
t[1, k_] := t[1, k] = If[k == nn, x, 0];
t[n_, k_] :=
  t[n, k] = ((1 - (t[n - 1, k - 1]*t[n - 1, k - 0]*
          t[n - 1, k + 1]))*(1 -
        t[n - 1, k - 1]*
         t[n - 1,
          k - 0]*(1 - t[n - 1, k + 1]))*(1 - (t[n - 1,
           k - 1]*(1 - t[n - 1, k - 0])*
          t[n - 1,
           k + 1]))*(1 - ((1 - t[n - 1, k - 1])*(1 -
            t[n - 1, k - 0])*(1 - t[n - 1, k + 1])))) + ((t[n - 1,
         k - 1]*(1 - t[n - 1, k - 0])*(1 - t[n - 1, k + 1]))*((1 -
          t[n - 1, k - 1])*t[n - 1, k - 0]*
        t[n - 1, k + 1])*((1 - t[n - 1, k - 1])*
        t[n - 1,
         k - 0]*(1 - t[n - 1, k + 1]))*((1 - t[n - 1, k - 1])*(1 -
          t[n - 1, k - 0])*t[n - 1, k + 1]));
TableForm[Table[Table[Expand[t[n, k]], {k, 1, 2*nn}], {n, 1, nn}]]
(*Mats Granvik,Dec 07 2019*)
(*end*)


(*start*)
Clear[t, n, k, x];
nn = 9;
t[1, k_] := t[1, k] = If[k == nn, x, 0];
t[n_, k_] :=
  t[n, k] =
   t[-1 + n, -1 + k] +
    t[-1 + n, 0 + k]*y + (1 + t[-1 + n, 0 + k]) t[-1 + n, +1 + k];
TableForm[Table[Expand[t[n, nn]], {n, 1, nn}]];
TableForm[Table[CoefficientList[Expand[t[n, nn]]/x, x], {n, 1, nn}]]
Table[Sum[
  Binomial[n, k]*Binomial[n - k, k]*Binomial[n - 2*k, 2*k], {k, 0,
   n}], {n, 0, 12}]
(*end*)

From Wikipedia 1.1.2020:

(*start*)
Clear[t, n, k, x];
nn = 200;
t[1, k_] := t[1, k] = If[k == nn, 0, 1];
t[n_, k_] :=
  t[n, k] =
   Mod[t[-1 + n, -1 + k] + 1 + t[-1 + n, 0 + k] t[-1 + n, +1 + k], 2];
ArrayPlot[Table[Table[1 - t[n, k], {k, 1, 2*nn}], {n, 1, nn}]]
(*end*)

Binomials found in terms of:
(*start*)
(*Some of the binomial coefficients in front of terms*)
Clear[t, n, k, x];
nn = 16;

t[1, k_] := t[1, k] = If[k == nn, x, 0];
t[n_, k_] :=
  t[n, k] =
   t[-1 + n, -1 + k] +
    t[-1 + n, 0 + k] + (1 + t[-1 + n, 0 + k]) t[-1 + n, +1 + k];

Do[
 Print[TableForm[Table[Part[t[n, nn], i], {n, 1, nn}]]], {i, 1, nn}]
(*end*)