Clear[t, n, k, a, b, x, y, b1, at, T, AT, at];
nn = 200;
x = 2 + 3 I;
y = 5 + 7*I;
T[n_, 1] = 1;
T[n_, k_] :=
  T[n, k] =
   If[n >= k,
    x*Sum[T[n - i, k - 1], {i, 1, n - 1}] +
     y*Sum[T[n - i, k], {i, 1, n - 1}], 0];
a = Table[Table[Expand[T[n, k]], {k, 1, nn}], {n, 1, nn}];
(*TableForm[a]*)
AT = Total[Transpose[a]];
AT[[1 ;; 20]];
Table[Limit[((s + 1)^(n - 1) + s - 1)/s, s -> y + 1], {n, 1, 12}];
c = Product[1/(1 - 1/AT[[i]]), {i, 2, nn}];
N[c, 12]

Clear[t];
t[n_, 1] = 1;
t[n_, k_] :=
  t[n, k] =
   If[n >= k,
    x*Sum[t[n - i, k - 1], {i, 1, k - 1}] +
     y*Sum[t[n - i, k], {i, 1, k - 1}], 0];
a = Table[Table[Expand[t[n, k]], {k, 1, nn}], {n, 1, nn}];
aa = Table[
   Total[Table[t[n - k + 1, k], {k, 1, nn}]], {n, 1 + 1, nn + 1}];
(*TableForm[a]*)
at = Total[Transpose[a]];
at[[1 ;; 20]];
N[AT[[nn]]/at[[nn]], 12]