Algorithm (Rolling Hash)

1. Brute force compare the hash of all substrings.
2. The hash could be computed efficiently using rolling hash technique.
3. Total complexity is $O(n^{2})$ where $n$ is the length of the input string.

Code

namespace math::field {

template <typename T>
T inverse(T a, T m) {
  T u = 0, v = 1;
  while (a != 0) {
    T t = m / a;
    m -= t * a; swap(a, m);
    u -= t * v; swap(u, v);
  }
  assert(m == 1);
  return u;
}

template <typename T>
class Z {
 public:
  using value_type = typename T::value_type;
  constexpr static value_type mod() { return T::value; }
  // constructors 
  constexpr Z() : value() {}

  template <typename U> Z(const U& x) { value = normalize(x); }
  template <typename U> static value_type normalize(const U& x) {
    if (0 <= x and x < mod()) return value_type(x);
    else if (x >= mod())      return value_type(x % mod());
    else                      return value_type(x + mod());
  }

  Z& operator+=(const Z& other) { if ((value += other.value) >= mod()) value -= mod(); return *this; }
  Z& operator-=(const Z& other) { if ((value -= other.value) < 0) value += mod(); return *this; }
  Z& operator++() { return *this += 1; }
  Z& operator--() { return *this -= 1; }
  Z operator++(int) { Z result(*this); *this += 1; return result; }
  Z operator--(int) { Z result(*this); *this -= 1; return result; }
  Z operator-() const { return Z(-value); }
  Z& operator*=(const Z& rhs) { value = (value % mod() * rhs.value % mod()) % mod(); return *this; }
  Z& operator/=(const Z& other) { return *this *= Z(inverse(other.value, mod())); }
  const value_type& operator()() const { return value; }

  template <typename U> explicit operator U() const { return static_cast<U>(value); }
  template <typename U> Z& operator+=(const U& other) { return *this += Z(other); }
  template <typename U> Z& operator-=(const U& other) { return *this -= Z(other); }  
  template <typename U> friend const Z<U>& abs(const Z<U>& v) { return v; }
  template <typename U> friend bool operator==(const Z<U>& lhs, const Z<U>& rhs);
  template <typename U> friend bool operator<(const Z<U>& lhs, const Z<U>& rhs);
  template <typename U> friend std::istream& operator>>(std::istream& stream, Z<U>& number);

 private:
  value_type value;
};

template <typename T> bool operator==(const Z<T>& lhs, const Z<T>& rhs) { return lhs.value == rhs.value; }
template <typename T, typename U> bool operator==(const Z<T>& lhs, U rhs) { return lhs == Z<T>(rhs); }
template <typename T, typename U> bool operator==(U lhs, const Z<T>& rhs) { return Z<T>(lhs) == rhs; }

template <typename T> bool operator!=(const Z<T>& lhs, const Z<T>& rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(const Z<T>& lhs, U rhs) { return !(lhs == rhs); }
template <typename T, typename U> bool operator!=(U lhs, const Z<T>& rhs) { return !(lhs == rhs); }

template <typename T> bool operator<(const Z<T>& lhs, const Z<T>& rhs) { return lhs.value < rhs.value; }

template <typename T> Z<T> operator+(const Z<T>& lhs, const Z<T>& rhs) { return Z<T>(lhs) += rhs; }
template <typename T, typename U> Z<T> operator+(const Z<T>& lhs, U rhs) { return Z<T>(lhs) += rhs; }
template <typename T, typename U> Z<T> operator+(U lhs, const Z<T>& rhs) { return Z<T>(lhs) += rhs; }

template <typename T> Z<T> operator-(const Z<T>& lhs, const Z<T>& rhs) { return Z<T>(lhs) -= rhs; }
template <typename T, typename U> Z<T> operator-(const Z<T>& lhs, U rhs) { return Z<T>(lhs) -= rhs; }
template <typename T, typename U> Z<T> operator-(U lhs, const Z<T>& rhs) { return Z<T>(lhs) -= rhs; }

template <typename T> Z<T> operator*(const Z<T>& lhs, const Z<T>& rhs) { return Z<T>(lhs) *= rhs; }
template <typename T, typename U> Z<T> operator*(const Z<T>& lhs, U rhs) { return Z<T>(lhs) *= rhs; }
template <typename T, typename U> Z<T> operator*(U lhs, const Z<T>& rhs) { return Z<T>(lhs) *= rhs; }

template <typename T> Z<T> operator/(const Z<T>& lhs, const Z<T>& rhs) { return Z<T>(lhs) /= rhs; }
template <typename T, typename U> Z<T> operator/(const Z<T>& lhs, U rhs) { return Z<T>(lhs) /= rhs; }
template <typename T, typename U> Z<T> operator/(U lhs, const Z<T>& rhs) { return Z<T>(lhs) /= rhs; }

// using namespace std;;




template<typename T, typename U>
Z<T> power(const Z<T>& a, const U& b) {
  assert(b >= 0);
  Z<T> x = a, res = 1;
  U p = b;
  while (p > 0) {
    if (p & 1) res *= x;
    x *= x;
    p >>= 1;
  }
  return res;
}

template <typename T>
bool is_zero(const Z<T>& number) {
  return number() == 0;
}

template <typename T>
string to_string(const Z<T>& number) {
  return to_string(number());
}

template <typename T>
std::ostream& operator<<(std::ostream& stream, const Z<T>& number) {
  return stream << number();
}

template <typename T>
std::istream& operator>>(std::istream& stream, Z<T>& number) {
  typename common_type<typename Z<T>::value_type, long long>::type x;
  stream >> x;
  number.value = Z<T>::normalize(x);
  return stream;
}
} // end of namespace 

template<typename T> struct std::hash<math::field::Z<T>> {
  std::size_t operator()(math::field::Z<T> const& z) const noexcept {
    return std::hash<typename T::value_type>{}(z());
  }
};



template <class F>
struct recursive {
  F f;
  template <class... Ts>
  decltype(auto) operator()(Ts&&... ts)  const { return f(std::ref(*this), std::forward<Ts>(ts)...); }

  template <class... Ts>
  decltype(auto) operator()(Ts&&... ts)  { return f(std::ref(*this), std::forward<Ts>(ts)...); }
};

template <class F> recursive(F) -> recursive<F>;
auto const rec = [](auto f){ return recursive{std::move(f)}; };




class Solution {
 public:
  int distinctEchoSubstrings(string text) {

    typedef std::integral_constant<long long, (long long)(1e9+7)> mod;
    using math::field::Z;

    const auto N = size(text);

    static const auto power = [&](vector<Z<mod>> self = {}) {
      self.resize(2000, 1);
      for (int i = 1; i < 2000; ++i) 
        self[i] = self[i - 1] * 256;
      return self;
    }();


    auto ascii = [](char ch) { return int(ch); };

    const auto prefix_rolling_hash = [&](vector<Z<mod>> self = {}) {
      self.resize(N);
      auto go = rec([&](auto&&go, int i) -> void {
        if (i == size(text))
          return;
        else if (i == 0) 
          (self[i] = ascii(text[i]), go(i + 1));
        else
          (self[i] = 256 * self[i - 1] + ascii(text[i]), go(i + 1));
      });
      return (go(0), self);
    }();

    auto substr_rolling_hash = [&](int i, int j) {
      if (i == 0)
        return prefix_rolling_hash[j];
      else 
        return prefix_rolling_hash[j] - prefix_rolling_hash[i - 1] * power[j - i + 1];
    };

    const auto solution = [&](unordered_set<Z<mod>> acc = {}) {
      auto go = rec([&](auto&&go, int i, int len) -> void {
        if (i + 2 * len - 1 >= N)
          return;
        else if (substr_rolling_hash(i, i + len - 1) == substr_rolling_hash(i + len, i + 2 *len - 1)) 
          (acc.emplace(substr_rolling_hash(i, i + len - 1)), go(i + 1, len));
        else
          go(i + 1, len);
      });

      for (int len = 1; len <= N / 2; ++len)
        go(0, len);

      return size(acc);
    }();

    return solution;

  }
};