Induced Subgraph Let $G = (V, E)$, be a connected graph with $V$ vertices and $E$ edges. We define $H(x)$ as the number of vertex induced subgraphs with $x$ edges in it.
As per definition, $\sum_{i=0}^{i=m} H(i) = 2^V$. Assume we consider null graph as also one of the induced subgraphs. 
For example, let $G(V, E) = \{(1, 2), (2, 3), (1, 3)\}$, where $V = \{1, 2, 3\}$, then $H(0) = 4$, $H(1) = 3$, $H(2) = 0$, $H(3) = 1$.
Also, suppose we write the polynomial $P(x) = \sum_{i=0}^{i=m} H(i) * x^i$, then, what can we say about $P(x)$ in terms of $V$ and $E$? This is because it can help us solve the above algorithm efficiently by differentiating the polynomial $E$ times, putting $x = 0$, every time and getting the corresponding coefficient.
Is there an efficient algorithm for the same?
Thanks for your help in advance. 
Update : Thanks to @Dap, my problem is solved. But I was curious to know if anyone can help with ideas to find $\sum_{i=0}^{i=m} {i}^{k} H(i)$ for fixed $k$ as for variable $k$, the solution will be '#-$P-hard$' as @Dap suggested in one of the comments. For example, what could be possible approaches for say $k = 2, 3, 4$. I found out that for $k = 0$, we have $\sum_{i=0}^{i=m} {i}^{0} H(i) = 2^V$, and for $k = 1$, we have $\sum_{i=0}^{i=m} {i}^{1} H(i) = E * 2^V$.
Update 2: The modified problem based on summation $\sum_{i=0}^{i=m} {i}^{k} H(i)$ is also solved now, thanks to @Dap.
 A: I will just discuss the computational problem of computing $P(x)$ from $(V,E)$, with $x$ fixed.
$P(x)$ can be put in the more general context of a "two spin model" partition function,
$$P(x)=\sum_{\sigma:V\to\{0,1\}}\prod_{ij\in E}W_{\sigma(i)\sigma(j)}$$
where
$$W=\begin{pmatrix}1&1\\1&x\end{pmatrix}$$
As well as the case $P(1)=2^V$ you mentioned, there is a polynomial time algorithm to compute $P(-1)$ exactly, and the other cases are NP-hard to compute exactly (in fact "#P-hard"), see A complexity dichotomy for partition functions with mixed signs. Note $P(0)$ is the number of independent sets.
For $x>1$ there is an FPRAS to approximate $P(x)$, whereas for $0\leq x<1$ it's NP-hard to approximate. See section 3.3.2 of this survey:  http://drops.dagstuhl.de/opus/volltexte/2017/6965/
(its Theorem 14 doesn't apply directly to $W$, but if you chase the reference [53] you'll see the NP-hardness is proved even for $\Delta$-regular graphs, and with $\beta=\gamma=\sqrt x$ and $\lambda=\sqrt x^\Delta$ you can show that the partition function comes out the same up to an easily-computed factor.)
