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The Secundus U(2,F)F
The series around U0:
| The secundus is symmetric with regard to U0. It can be derived from the primus by taking the subsequent differences between Un+1 and Un-1 of that series (with U0 = u1 - u-1 = 2).
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The secundus coefficients matrix
The degree of a polynome the same as 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.
| U1: | 1 | ||||||||||||
| U2: | 1 | -2 | |||||||||||
| U3: | 1 | -3 | |||||||||||
| U4: | 1 | -4 | 2 | ||||||||||
| U5: | 1 | -5 | 5 | ||||||||||
| U6: | 1 | -6 | 9 | -2 | |||||||||
| U7: | 1 | -7 | 14 | -7 | |||||||||
| U8: | 1 | -8 | 20 | -16 | 2 | ||||||||
| U9: | 1 | -9 | 27 | -30 | 9 | ||||||||
| U10: | 1 | -10 | 35 | -50 | 25 | -2 | |||||||
| U11: | 1 | -11 | 44 | -77 | 55 | -11 | |||||||
| U12: | 1 | -12 | 54 | -112 | 105 | -36 | 2 | ||||||
| U13: | 1 | -13 | 65 | -156 | 182 | -91 | 13 | ||||||
| U14: | 1 | -14 | 77 | -210 | 294 | -196 | 49 | -2 | |||||
| U15: | 1 | -15 | 90 | -275 | 450 | -378 | 140 | -15 | |||||
| U16: | 1 | -16 | 104 | -352 | 660 | -672 | 336 | -64 | 2 | ||||
| U17: | 1 | -17 | 119 | -442 | 935 | -1122 | 714 | -204 | 17 | ||||
| U18: | 1 | -18 | 135 | -546 | 1287 | -1782 | 1386 | -540 | 81 | -2 | |||
| U19: | 1 | -19 | 152 | -665 | 1729 | -2717 | 2508 | -1254 | 285 | -19 | |||
| U20: | 1 | -20 | 170 | -800 | 2275 | -4004 | 4290 | -2640 | 825 | -100 | 2 | ||
| U21: | 1 | -21 | 189 | -952 | 2940 | -5733 | 7007 | -5148 | 2079 | -385 | 21 | ||
| U22: | 1 | -22 | 209 | -1122 | 3740 | -8008 | 11011 | -9438 | 4719 | -1210 | 121 | -2 | |
| U23: | 1 | -23 | 230 | -1311 | 4692 | -10948 | 16744 | -16445 | 9867 | -3289 | 506 | -23 | |
| U24: | 1 | -24 | 252 | -1520 | 5814 | -14688 | 24752 | -27456 | 19305 | -8008 | 1716 | -144 | 2 |
| U25: | 1 | -25 | 275 | -1750 | 7125 | -19380 | 35700 | -44200 | 35750 | -17875 | 5005 | -650 | 25 |