RosettaCodeData/Task/Forward-difference/00DESCRIPTION

;Task:
Provide code that produces a list of numbers which is the &nbsp; <big>n<sup>th</sup></big> &nbsp;order forward difference, given a non-negative integer (specifying the order) and a list of numbers.


The first-order forward difference of a list of numbers &nbsp; <big>'''A'''</big> &nbsp; is a new list &nbsp; <big>'''B'''</big>, &nbsp; where &nbsp; <big><b>B</b><sub>n</sub> = <b>A</b><sub>n+1</sub> - <b>A</b><sub>n</sub></big>.

List &nbsp; <big>'''B'''</big> &nbsp; should have one fewer element as a result.

The second-order forward difference of &nbsp; <big>'''A'''</big> &nbsp; will be:
<pre>
tdefmodule Diff do
	def forward(arr,i\\1) do
		forward(arr,[],i)
	end

	def forward([_|[]],diffs,i) do
		if i == 1 do
			IO.inspect diffs
		else
			forward(diffs,[],i-1)
		end
	end

	def forward([val1|[val2|vals]],diffs,i) do
		forward([val2|vals],diffs++[val2-val1],i)
	end
end
</pre>
The same as the first-order forward difference of &nbsp; <big>'''B'''</big>.

That new list will have two fewer elements than &nbsp; <big>'''A'''</big> &nbsp; and one less than &nbsp; <big>'''B'''</big>.

The goal of this task is to repeat this process up to the desired order.

For a more formal description, see the related &nbsp; [http://mathworld.wolfram.com/ForwardDifference.html Mathworld article].


;Algorithmic options:
* Iterate through all previous forward differences and re-calculate a new array each time.
* Use this formula (from [[wp:Forward difference|Wikipedia]]):
<big><big>
::: <math>\Delta^n [f](x)= \sum_{k=0}^n {n \choose k} (-1)^{n-k} f(x+k)</math>
</big></big>
::: ([[Pascal's Triangle]] &nbsp; may be useful for this option.)
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