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The Primus U(0,1)F
The series around U_{0}:
 The primus is semisymmetric with regard to U_{0}. The other three of the fabfour can be derived from it by taking subsequent sums and differences of its terms (quartus and tertius) or subsequent differences between U_{n+1} and U_{n1} (secundus). Its coefficients matrix is, like the sigma repeated matrix, another representation of Pascal's triangle.

The following theorems link terms around U_{n} with terms at twice the index value. This is the series' development 'from the belly'. Theorem 2 is important as well as elegant: it shows that the primus on odd indices equals (tertius)*(quartus) while on even indices it equals (primus)*(secundus). Thus, factorization of the primus effectively boils down to the factorization of the other three.  
U_{2n1} =  (U_{n}U_{n1})(U_{n}+U_{n1})  Theorem 2.1 
U_{2n} =  U_{n}(U_{n+1}U_{n1})  Theorem 2.2 
U_{2n1} =  U_{n}(U_{n}U_{n2})1  Theorem 3.1 
U_{2n} =  (U_{n+1}+U_{n})(U_{n}U_{n1})1  Theorem 3.2 
The primus coefficients matrix
The degree of a polynome is one less than its index.
Exponents decrease with steps of 2.
Note that each column displays the subsequent differences in the next one, or, to put it another way, each column displays the partial sums of the previous one.
Pascal's triangle appears starting at the top and looking right and down: 1  11  121  1331  14641 etc. To simplify the comparison, the main diagonal has been indicated.
Like Pascal's triangle, this matrix is the sigma repeated matrix revisited.
U_{1}:  1  
U_{2}:  1  
U_{3}:  1  1  
U_{4}:  1  2  
U_{5}:  1  3  1  
U_{6}:  1  4  3  
U_{7}:  1  5  6  1  
U_{8}:  1  6  10  4  
U_{9}:  1  7  15  10  1  
U_{10}:  1  8  21  20  5  
U_{11}:  1  9  28  35  15  1  
U_{12}:  1  10  36  56  35  6  
U_{13}:  1  11  45  84  70  21  1  
U_{14}:  1  12  55  120  126  56  7  
U_{15}:  1  13  66  165  210  126  28  1  
U_{16}:  1  14  78  220  330  252  84  8  
U_{17}:  1  15  91  286  495  462  210  36  1  
U_{18}:  1  16  105  364  715  792  462  120  9  
U_{19}:  1  17  120  455  1001  1287  924  330  45  1  
U_{20}:  1  18  136  560  1365  2002  1716  792  165  10  
U_{21}:  1  19  153  680  1820  3003  3003  1716  495  55  1  
U_{22}:  1  20  171  816  2380  4368  5005  3432  1287  220  11  
U_{23}:  1  21  190  969  3060  6188  8008  6435  3003  715  66  1  
U_{24}:  1  22  210  1140  3876  8568  12376  11440  6435  2002  286  12  
U_{25}:  1  23  231  1330  4845  11628  18564  19448  12870  5005  1001  78  1 
U_{26}:  1  24  253  1540  5985  15504  27132  31824  24310  11440  3003  364  13 