Hypergeometric function and divisors

Clear[t, n, k, x];
nn = 100;
x = 1;
t[n_, 1] = 1;
t[n_, k_] :=
 t[n, k] =
  If[And[k > 1, n >= k],
   Sum[t[n - i, k - 1]/x - t[n - i, k], {i, 1, k - 1}]/k, 0]
Monitor[Table[Sum[t[n, k], {k, 1, n}], {n, 1, nn}];, n]
N[%, 30]
N[Hypergeometric2F1[1/2, 1/2, 3/2, (2*x - 1)/(x^2)], 30]

x = 1;
Hypergeometric2F1[1/2, 1/2, 3/2, (2*x - 1)/(x^2)]