t-curve Interpolation

Suppose you have a group of points of a one-to-one function like so:
[1,2], [3,4],[4,6],[6,10]
I found that you can use what I'm calling a t-curve that for four points looks like:

Then you simply solve this f through those points in Maple it is done like this:

Then plot your curve with these coefficients:

See it goes smoothly through the given points! Another example:
Points:
[2,4],[3,2],[4,3]

One more continuing from the last curve:
[2,4],[3,2],[4,3], [5,1]
The same first 3 points as the last but with a fourth point [5,1]...

I'm pretty sure there is always a solution but not positive...
The main drawback is the coefficients can get large, but it has a nice form I think...

**Addendum**
This formulation might be better as it seems to keep the coefficients more reasonably sized and nice fractions, the ith term is multiplied by i!: