Creating a summation table

Question

Let's say I have a list of values, for example {3,7,8,10}. Now I want to make a function that creates a table T in the following way:

$\begin{matrix} 3&7&8&10\\ &10&15&18\\ &&25&33\\ &&&58\\ \end{matrix}$

So we're adding up: 3+7=10, 7+8=15 etc.

Another complicating factor is that I want T[[2,2]]=10 (not T[[2,1]]); similarly, T[[4,4]]=58 etc.

BACKGROUND: In the end I am trying to make a function for Romberg Integration.

Answer 1

lst = {3, 7, 8, 10};

nl = NestList[MovingMap[Total, #, 1]&, lst, Length[lst] - 1]

{{3,7,8,10},{10,15,18},{25,33},{58}}

TeXForm @ MatrixForm @ PadLeft[nl, Automatic, ""]

$\left( \begin{array}{cccc} 3 & 7 & 8 & 10 \\ \text{} & 10 & 15 & 18 \\ \text{} & \text{} & 25 & 33 \\ \text{} & \text{} & \text{} & 58 \\ \end{array} \right)$

Alternatively, you can also use

Total /@ Partition[#, 2, 1] & (* or *)

Partition[#, 2, 1, {1, -1}, {}, Plus] &

in place of MovingMap[Total, #, 1]& above.

To get a 4X4 matrix, you can use PadLeft (with default padding 0):

nl2 = PadLeft[nl]

{{3, 7, 8, 10}, {0, 10, 15, 18}, {0, 0, 25, 33}, {0, 0, 0, 58}}

nl2[[4, 4]]

58

TeXForm @ MatrixForm @ nl2

$\left( \begin{array}{cccc} 3 & 7 & 8 & 10 \\ 0 & 10 & 15 & 18 \\ 0 & 0 & 25 & 33 \\ 0 & 0 & 0 & 58 \\ \end{array} \right)$