Background
Consider the following representation of a function:
$$ f(x) = a_0 + a_1 e^{x} + a_2 e^{x^2} + a_3 e^{x^3} + \dots $$
Then,
$$f(0) = \sum_{i} a_i$$
Similarly,
$$ f'(x) = a_1 e^x + 2 a_2 x e^{x^2} + 3 a_3 x^2 e^{x^3} + \dots $$
$$\implies f'(0) = a_1$$
Or upon, again differentiating the equation we could have got:
$$ f''(x) = a_1 e^x + 2a_2 e^{x^2} + 4 a_2 x^2 e^{x^2} + \dots$$
Again, putting $x = 0$:
$$ f''(0) = a_1 + 2a_2 $$
Question
Is there some nice formula for $f'' ^n (0) $ ?