I have a list of numbers and I want to express each one as an integer linear combination of $2^{(2^k)}$ powers.

Some elements of the list are 

```
(3748, 64594, 16700300, 68685479824, 4722365919717929558384, ...)

```

The first two elements can be expressed for example as

$$ 3748 = 2^2 - 6\cdot2^4 + 15\cdot2^8 $$

and

$$ 16700300= - 2^2 + 9\cdot2^4 - 45\cdot2^8 + 255\cdot2^{16}, $$

where the exponents $2$, $4$, $8$ and $16$ are all powers of 2.

Can you help me write a program which yields such coefficients?
the answers must be of the form ...+-+-+-...

```
3748 ===> {1, -6, 15}

16700300 ===> {-1, 9, -45, 255}

```

I also realised that if the prime factors of the number are not raised to any power then you can find the coefficients here (i.e. 2\*3\*5\*7 has n=4)

```
  n= number of distinct primes
 Table[(-1)^(n - k) Binomial[n, k], {k, 0, n}]

```

Is there a formula for factors raised to powers?