We can form a series that is somewhat easier to work with by noting that $ \ \frac{d}{dx} \ [e^{x^3}] \ = \ 3x^2 · e^{x^3} \ \ , $ so that we produce the anti-derivative series
$$ G(x) \ \ = \ \ \frac13 \ e^{x^3} \ \ = \ \frac13 \ \sum_{n \ = \ 0}^{\infty} \ \frac{1}{n!} \ (x^3)^n \ \ = \ \ \frac13 \ \sum_{n \ = \ 0}^{\infty} \ \frac{x^{3n}}{n!} \ \ , $$
with $ \ G'(x) \ = \ g(x) \ = \ x^2 e^{x^3} \ . $ The series for $ \ G(x) \ $ is non-zero only for every third term, that is to say,
$$ G^{(n)}(0) \ \ = \ \ \left\{ \begin{array}{rcl} a_{n} \ \ , & \ \text{for} \ n = 3k \ \ , \ \ k \ \ \text{a non-negative integer} \\ 0 \ \ , & \text{otherwise} \end{array} \right. \ \ . $$
Since $ \ g^{(40)}(0) \ = \ G^{(41)}(0) \ \ , $ and $ \ 41 \ $ is not a multiple of $ \ 3 \ \ , $ we can simply conclude that $ \ g^{(40)}(0) \ = \ 0 \ \ . $ (Had the problem asked for $ \ g^{(41)}(0) \ \ \text{or} \ \ g^{(50)}(0) \ \ $ (as appears in one of the answers -- the original version of the post?), we would have more work to do, as discussed below.
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Another way we can approach the problem is to make use of the "higher-derivatives" version of the Product Rule,
$$ \frac{d^n}{dx^n} \ [f_1(x) · f_2(x)] \ \ = \ \ \sum_{k \ = \ 0}^{n} \ \binom{n}{k} \ f_1^{(k)}(x) · f_2^{(n-k)}(x) \ \ , $$
which behaves like a binomial expansion. This will be of help to us if we define, as we did above, $ \ G(x) \ = \ \frac{1}{3} · e^{x^3} \ = \ \frac{1}{3} · e^u \ = \ y \ \ , $ so that $ \ g(x) \ = \ G'(x) \ = \ u' · y \ \ . $ We will then set $ \ f_1 = u' \ $ and $ \ f_2 = y \ \ $ in making our calculations.
This may look daunting for the derivative we wish to determine, but there are features of the function that lead to considerable simplification. We have $ \ G(0) = \frac13 \ $ and $ \ u = x^3 \ \Rightarrow \ u' = 3x^2 \ , \ u'' = 6x \ , \ u''' = 6 \ \ , $ and $ \ u^{(k)} = 0 \ $ for $ \ k \ge 4 \ \ . $ What this tells us is that the number of terms in the expansion for successive derivatives is quite limited. Moreover, since we are evaluating the derivatives of $ \ g(x) \ $ at $ \ x = 0 \ $ , the only non-zero value for the $ \ u^{(n)} $ 's is $ \ u''' = 6 \ \ . $
With this in mind, we see that
$ g(0) \ = \ G'(0) \ = \ y'(0) \ = \ 0 · \frac13 \ = \ 0 \ \ , $
$ g'(x) \ = \ G''(x) \ = \ y'' \ = \ u''· y \ + \ u'· y' \ \ \Rightarrow \ \ g'(0) \ = \ 0 · \frac13 \ + \ 0 · 0 \ = \ 0 \ \ , $
$ g''(x) \ = \ y''' \ = \ u'''· y \ + \ 2 · u''· y' \ + \ u'·y'' \ \ \Rightarrow \ \ g''(0) \ = \ 6 · \frac13 \ + \ 2 · 0 · 0 \ + \ 0·0 \ = \ 2 \ \ . $
Here, we attain our first non-zero derivative with $ \ G'''(0) \ = \ g''(0) \ \ . $ With the next derivative, our expressions begin to include derivatives of $ \ u \ $ which are simply zero.
$ g'''(x) \ = \ y^{(4)} \ = \ u^{(4)}· y \ + \ 3 · u'''· y' \ + \ 3·u''·y'' \ + \ u'·y''' $
$ \Rightarrow \ \ g'''(0) \ = \ 0 · \frac13 \ + \ 3 · \frac13 · 0 \ + \ 3·0·0 \ + \ 0·6 \ = \ 0 \ \ , $
$ g^{(4)}(x) \ = \ y^{(5)} \ = \ u^{(5)}· y \ + \ 4 · u^{(4)}· y' \ + \ 6·u'''·y'' \ + \ 4·u''·y''' \ + \ u'·y^{(4)} $
$ \Rightarrow \ \ g^{(4)}(0) \ = \ 0 \ + \ 0 \ + \ 6 · \frac13 · 0 \ + \ 4·0·2 \ + \ 0 \ = \ 0 \ \ , $
and
$ g^{(5)}(x) \ = \ y^{(6)} \ = \ u^{(6)}· y \ + \ 5 · u^{(5)}· y' \ + \ 10·u^{(4)}·y'' \ + \ 10·u'''·y''' \ + \ 5·u''·y^{(4)} \ + \ u'·y^{(5)} $
$ \Rightarrow \ \ g^{(5)}(0) \ = \ 0 \ + \ 0 \ + \ 0 \ + \ 10·6·2 \ + \ 0 \ + \ 0 \ = \ \binom{5}{3}·6·2 \ = \ 120 \ \ . $
What becomes clear from this examples is that the non-zero derivatives for non-negative integer $ \ k \ $ are given by the recursion relation
$$ g^{(3k + 2)}(0) \ = \ y^{(3k + 3)}(0) \ = \ \binom{3k+2}{3k}·u'''·y^{(3k)}(0) \ = \ \binom{3k+2}{3k}·6·y^{(3k)}(0) \ \ , \ \ y(0) \ = \ \frac13 \ \ . $$
Hence,
$$ g^{(8)}(0) \ = \ y^{(9)}(0) \ = \ \binom{8}{6}·6·y^{(6)}(0) \ = \ \binom{8}{6}·6· \left[\binom{5}{3}·6·2 \right] \ = \ 20,160 \ \ , $$
$$ g^{(11)}(0) \ = \ y^{(12)}(0) \ = \ \binom{11}{9}·6·y^{(9)}(0) \ = \ 55·6·20,160 \ = \ 6,652,800 \ \ , $$
and so forth. We observe that while $ \ g^{(41)}(0) \ = \ y^{(42)}(0) \ $ would be enormous, $ \ g^{(40)}(0) \ = \ y^{(41)}(0) \ $ is not in this set of derivatives, and so is equal to zero.
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We might like to have something a bit simpler to work with than the recursion relation for the non-zero derivatives. If we write out the last example in full, we obtain
$$ g^{(11)}(0) \ \ = \ \ \binom{11}{9}·6·\binom{8}{6}·6· \binom{5}{3}·6·\binom{2}{0}·6·\frac13 $$ $$ = \ \ \frac{11·10·9·8·7·6·5·4·3·2·1}{2 \ · \ 9 \ · \ 2 \ · \ 6 \ · \ 2 \ · \ 3 \ · \ 2} \ · \ 6^3 \ · \ 2 \ \ = \ \ \frac{11!}{2^3 \ · \ 3^3 \ · \ (3·2·1) \ · \ 2} \ · \ 6^3 \ · \ 2 \ \ = \ \ \frac{11!}{3!} \ \ . $$
The general calculation yields $ g^{(3k + 2)}(0) \ = \ \frac{(3k+2)!}{k!} \ \ , $ for integer $ \ k \ge 0 \ $ ; this completes the derivation characterization given above
$$ G^{(n+3)}(0) \ \ = \ \ \left\{ \begin{array}{rcl} \frac{(3k+2)!}{k!} \ \ , & \ \text{for} \ n = 3k \ \ , \ \ k \ \ \text{a non-negative integer} \\ 0 \ \ , & \text{otherwise} \end{array} \right. \ \ . $$
Inserting this into the general term for a Maclaurin series, we then produce
$$ g(x) \ \ = \ \ x^2 · e^{x^3} \ \ = \ \ \sum_{m \ = \ 0}^{\infty} \ \frac{g^{(m)}(0) }{m!} \ x^m $$
[removing the "zero terms"]
$$ = \ \ \sum_{k \ = \ 0}^{\infty} \ \frac{(3k+2)! \ / \ k! }{(3k+2)!} \ x^{3k+2} \ \ = \ \ \sum_{k \ = \ 0}^{\infty} \ \frac{x^{3k+2} }{k!} \ \ = \ \ x^2 \ + \ x^5 \ + \ \frac12 · x^8 \ + \ \frac16 · x^{11} \ + \ \ldots \ \ , $$
in agreement with the method of substitution into the series for $ \ e^u \ $ and Gary's comment.
Best Answer
I think that you are supposed to directly start with $$y=\sum_{n=0}^\infty a_n\,x^n$$ to make $$\sum_{n=0}^\infty n\,a_n\,x^{n-1}=\sum_{k=0}^\infty \frac {x^{k+1}}{k!}$$ Expand just a little $$a_1+2a_2x+3a_3 x^2+4a_4 x^3+\cdots=\frac x{1!}+\frac {x^2}{1!}+\frac {x^3}{2!}+\cdots$$
So, just looking, $a_0$ is undefined (this is normal; it will be the constant of integration.
$a_1=0$ (since no constant term in the rhs)
Now, for the other, I am sure that you see how they are formed.
So, as you wrote $$y=c+\sum_{n=2}^\infty a_n\,x^n$$
Now, shift the index to start at $0$ if you wish.