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This is a nice trick to speed up calculations into the high-index realm.

The speed with which a series can be developed can be accellerated by repeated application of the mapping:
• U(a,b)F U(a,U2)[F2-2]
This is a base-2 accelleration (because it uses U2). Let's look at U(1,1)5 - the terius of 5 - for an example:
 1 1 4 19 91 436 2089 10009 47956 229771 1100899 5274724 25272721 121088881 580171684 2779769539 13318676011 63813610516 305749376569 1464933272329 7018916985076 U0 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12 U13 U14 U15 U16 U17 U18 U19 U20

After the mapping U(1,1)5 U(1,4)23 we look at the latter:
 1 4 91 2089 47956 1100899 25272721 580171684 13318676011 305749376569 7018916985076 U0 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10

After the mapping U(1,4)23 U(1,91)527 we look at the latter:
 1 91 47956 25272721 13318676011 7018916985076 U0 U1 U2 U3 U4 U5

After the mapping U(1,91)527 U(1,47956)277727 we look at the latter:
 1 47956 13318676011 U0 U1 U2

This, then, is the general idea base-2. It zaps down the parent series with steps that are powers of 2: 2, 4, 8, 16, ...
We can do it base-3, but that's just like base-2 really. It employs the mapping:
• U(a,b)F U(a,U3)[F3-3F]
Applied repeatedly, it one zaps down the parent series with steps that are powers of 3: 3, 9, 27, 81, ...

Matrices of the Factor base-n
 Base-1 1 Base-2 1 -2 Base-3 1 -3 Base-4 1 -4 2 Base-5 1 -5 5 Base-6 1 -6 9 -2 Base-7 1 -7 14 -7 Base-8 1 -8 20 -16 2 Base-9 1 -9 27 -30 9 Base-10 1 -10 35 -50 25 -2 Base-11 1 -11 44 -77 55 -11 Base-12 1 -12 54 -112 105 -36 2 ...
Of course we need the coefficients matrix of the factor F up to 'base-n' so here is how to go about that:

In this matrix every column consists of the subsequent differences of the next column. Using this property the matrix can be extended indefinitely. The base defines the highest power of the factor F. Powers decrease with steps of 2.
The general term of the polynom base-n could be constructed, but considering the size of the factors that can be reached by extending the matrix, this would seem a bit premature.

 Base-1 1 Base-2 1 2 Base-3 1 2 +1 Base-4 1 2 Base-5 1 2 -1 -2 1 Base-6 1 2 -2 -4 1 2 Base-7 1 2 -3 -6 2 4 +1 Base-8 1 2 -4 -8 4 8 Base-9 1 2 -5 -10 7 14 -2 -4 1 Base-10 1 2 -6 -12 11 22 -6 -12 1 2 Base-11 1 2 -7 -14 16 32 -13 -26 3 6 +1 Base-12 1 2 -8 -16 22 44 -24 -48 9 18 ...
In a non-zero-series U(a,b,c)F the factor F follows the above matrix just as in a zero-series. The constant c develops as c*(matrix over 1), and that one looks like this:

Note that each column consists if the subsequent differences, not so much of the next column, but of the one next to that. Using this property the matrix can be extended indefinitely. The power of the factor F is one less than the base. Powers decrease with steps of 1.

I hate to work without examples, so let's have a look at U(1,1,2)3 and accelerate it base-5.
Here's the parent series:
 1 1 4 13 37 100 265 697 1828 4789 12541 32836 85969 225073 589252 1542685 4038805 10573732 27682393 72473449 189737956 496740421 1300483309 3404709508 8913645217 23336226145 61095033220 159948873517 U0 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12 U13 U14 U15 U16 U17 U18 U19 U20 U21 U22 U23 U24 U25 U26 U27
The matrix over a and b renders F5 - 5F3 + 5F.
The matrix over c renders c*(F4 + 2F3 - F2 - 2F + 1).
For F=3 and c=2 this adds up to a new factor 123 and a new constant 242 respectively.
It would seem that the mapping we're after is: U(1,1,2)3 U(1,100,242)123
After this mapping we look at the latter again:
 1 100 12541 1542685 189737956 23336226145 U0 U1 U2 U3 U4 U5

Bingo!
Let's repeat the mapping base-5.
For F=123 and c=242 this adds up to a new factor 28143753123 and a new constant 56287506242 respectively.
It would seem that the mapping we're after is: U(1,100,242)123 U(1,23336226145,56287506242)28143753123
After this mapping we look at the latter again:
 1 23336226145 656768987503665507076 U0 U1 U2