I was not able to figure out what you did in Wolfram|Alpha (a specific example is always helpful). Possibly this will give a start along the lines of what you want.
aa = {1, 81, 28, 13, 19, 57, 25, 87, 19, 44};
NestList[Differences, aa, 3]
(* {{1, 81, 28, 13, 19, 57, 25, 87, 19, 44}, {80, -53, -15, 6, 38, -32,
62, -68, 25}, {-133, 38, 21, 32, -70, 94, -130, 93}, {171, -17,
11, -102, 164, -224, 223}} *)
This variant will go all the way to the root, so to speak.
Most[FixedPointList[Differences, aa]]
(* Out[37]= {{1, 81, 28, 13, 19, 57, 25, 87, 19, 44}, {80, -53, -15, 6,
38, -32, 62, -68, 25}, {-133, 38, 21, 32, -70, 94, -130,
93}, {171, -17, 11, -102, 164, -224, 223}, {-188, 28, -113,
266, -388, 447}, {216, -141, 379, -654, 835}, {-357, 520, -1033,
1489}, {877, -1553, 2522}, {-2430, 4075}, {6505}, {}} *)