What is the method for finding the summation of a series whose $n^{th}$ difference between consecutive terms is in an AP?
For example, $$2,12,36,80,150,252...$$ Taking the first term difference we get another series as such - $10,24,44,70,102...$.
Taking it's term difference we get the series $14,20,26,32$ which is in an Arithmetic Progression.
Or, $$1,13,53,143,311,591,1023...$$
In this, the series formed by the $3^{rd}$ order difference is in AP.
Let $(a_k)_{k\geq 0}$ be a sequence which is an arithmetical progression of order $n$. We show the following is valid: \begin{align*} \color{blue}{\sum_{k=0}^qa_k =\sum_{j=0}^n\binom{q+1}{j+1}\Delta^j a_0 \qquad\qquad q\geq n}\tag{0} \end{align*}
We use the shift operator $E$, which applied at $a_k$ gives $E a_k=a_{k+1}$, $k\geq 0$ and we als need the forward difference operator $\Delta a_k=a_{k+1}-a_k$. The shift operator $E$ can be written in terms of the $\Delta$ operator as \begin{align*} E&=\Delta+I\tag{1} \end{align*} since \begin{align*} E a_k&=a_{k+1}\\ (\Delta+I)a_k&=\Delta a_k + I a_k=(a_{k+1}-a_k)+a_k=a_{k+1}\\ \end{align*} where $I a_k=a_k$ is the identity operator.
Let's take a look at OP's example: \begin{align*} (a_k)_{k\geq 0}=(2,12,36,80,150,252,\ldots)\tag{2} \end{align*} which is an arithmetical progression of order $3$. To better see what's going on we calculate the differences up to the fourth order. \begin{align*} \begin{array}{r|ccccccccccccc} a_k\ &\color{blue}{2}&&12&&36&&80&&150&&252&&\cdots\\ \Delta a_k\ &&\color{blue}{10}&&24&&44&&70&&102&&\cdots\\ \Delta^2 a_k\ &&&\color{blue}{14}&&20&&26&&32&&\cdots\\ \Delta^3 a_k\ &&&&\color{blue}{6}&&6&&6&&\cdots\\ \Delta^4 a_k\ &&&&&0&&0&&\cdots\\ \end{array} \end{align*}
We see the row $\Delta^3 a_k$ contains the constant $6$ only, indicating $(a_k)_{k\geq 0}$ is an arithmetical progression of order $3$. The left-most elements of $\Delta a_k^j, 0\leq j\leq 3$ are marked in blue. These are coefficients which will be relevant for the wanted formula.
We can write $E^m a_k=a_{k+m}$ with $m,k\geq 0$ in terms of the difference operator and obtain applying the binomial theorem \begin{align*} E^m&=(\Delta+I)^m=\sum_{j=0}^m\binom{m}{j}\Delta^j I^{m-j}\\ E^m a_k=a_{k+m}&=(\Delta+I)^ma_k =\sum_{j=0}^m\binom{m}{j}\Delta^j I^{m-j}a_k\\ &=\sum_{j=0}^m\binom{m}{j}\Delta^j a_k\tag{3}\\ \end{align*} Setting $k=0$ and $m=k$ in (3) we obtain a representation of $a_k$ in terms of the $j$-th differences $\Delta^j a_0$ of $a_0$ \begin{align*} a_k=\sum_{j=0}^{\min\{k,n\}}\binom{k}{j}\Delta^j a_0\qquad\qquad k\geq 0\tag{4} \end{align*} Note there are no more than $n+1$ terms in the right-hand side of (3), since $\Delta^j a_0=0$ if $j>n$.
We finally calculate a formula for the sum of the first $q$ terms $a_k$ from (4). To simplify things somewhat, we only consider the case $q\geq n$.
We obtain for $q\geq n$: \begin{align*} \color{blue}{\sum_{k=0}^q}\color{blue}{a_k}&=\sum_{k=0}^q \sum_{j=0}^n\binom{k}{j}\Delta^j a_0\tag{5}\\ &=\sum_{j=0}^n\sum_{k=0}^q \binom{k}{j}\Delta^j a_0\tag{6}\\ &=\sum_{j=0}^n\Delta^j a_0\sum_{k=j}^q \binom{k}{j}\tag{7}\\ &\,\,\color{blue}{=\sum_{j=0}^n\binom{q+1}{j+1}\Delta^j a_0}\tag{8}\\ \end{align*} and the claim (0) follows.
Comment:
In (5) we use formula (4).
In (6) we exchange the sums.
In (7) we factor out $\Delta^j a_0$ and set the lower index of the inner sum to $j$ since $\binom{k}{j}=0$ if $0\leq k<j$.
In (8) we apply the Hockey stick identity.
Now its time to harvest. We apply formula (8) to calculate for instance in OP's example the sum of the first $q+1=6$ elements. On the one hand we have from (2) \begin{align*} \sum_{k=0}^5a_k&=2+12+36+80+150+252\\ &\,\,\color{blue}{=532} \end{align*} on the other hand we obtain from (8) and the blue marked entries from the differences triangle: \begin{align*} \sum_{k=0}^5a_k&=\sum_{j=0}^3\binom{6}{j+1}\Delta^j a_0\\ &=\binom{6}{1}2+\binom{6}{2}10+\binom{6}{3}14+\binom{6}{4}6\\ &=12+150+280+90\\ &\,\,\color{blue}{=532} \end{align*}
Note that in the first calculation we always have to add $q+1$ terms in case of $q\geq n$, while in the second we have to sum up no more than $n+1$ terms, $n$ giving the order of the arithmetic progression.
Note: As preparatory work for this answer I went through section 23: Expansion into a Series of Binomial Coefficients of Calculus of finite Differences by C. Jordan.
Let $\sum_{i = 1}^{n} a_{i}$ be the series and let $b_{n} = a_{n + 1} - a_{n}$ be the sequence of differences of $a_{n}$. If the first term of the sequence is $a_{1}$ and $b_{n}$ is an arithmetic sequence with common difference $d$, then the formula for the sum of the series in terms of $a_{1}$, $b_{1}$ and $d$ is
$$\sum_{i = 1}^{n} a_{i} = na_{1} + n(n - 1)b_{1} + \frac{n(n - 1)(2n - 1)}{6}d$$
Proof
Since $b_{n}$ is an arithmetic progression, then each term $b_{i}$ can be expressed in terms of its first term $b_{1}$ and common difference $d$ as
$$b_{i} = b_{1} + (n - 1)d$$
By definition of $b_{n}$, we have
$$a_{i + 1} - a_{i} = b_{1} + (n - 1)d$$
Summing up all sides up to $n - 1$, we have
$$\sum_{i = 1}^{n - 1} a_{i + 1} - \sum_{i = 1}^{n - 1} a_{i} = (n - 1)b_{1} + {(n - 1)}^2d$$
The terms cancel in the left-hand side so that $a_{n} - a_{1}$ remains. Transposing $a_{1}$ to the right-hand side, we have
$$a_{n} = a_{1} + (n - 1)b_{1} + + {(n - 1)}^2d$$
At this point, we have derived the general term of the series $\sum_{i = 1}^{n} a_{i}$ in terms of $a_{1}$, $b_{1}$ and $d$. Summing up both sides up to $n$, we have
$$\sum_{i = 1}^{n} a_{i} = na_{1} + n(n - 1)b_{1} + \left(\sum_{i = 1}^{n} {(n - 1)}^2\right)d$$
From the sum of squares formula, we have
$$\sum_{i = 1}^{n} (n - 1)^2 = \frac{n(n - 1)(2n - 1)}{6}$$
Plugging this into the last result, we have
$$\sum_{i = 1}^{n} a_{i} = na_{1} + n(n - 1)b_{1} + \frac{n(n - 1)(2n - 1)}{6}d$$
and the proof is done.
