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Sigma repeated
We'll define   (Sigma repeated m over n) as follows:

for all natural n Definition 1.1
for all natural m and n Definition 1.2
Properties of Sigma repeated:
Property 1
Property 2
Property 3
Property 4
Property 5
Property 6

The Sigma repeated matrix
To the right:   indicated in darkblue - down: 'n' indicated by   in the second column.

The main diagonal has been indicated.
The matrix is symmetrical with regard to this diagonal.
The sub-diagonals are Pascal's triangle.
In yet another form it emerges as the coefficients matrix of the primus.
It also played a crucial role in determining the general term of U(a,b,c)F.

0
n		 1
n		 2
n		 3
n		 4
n		 5
n		 6
n		 7
n		 8
n		 9
n		 10
n		 11
n		 12
n		 13
n		 14
n		 15
n		 16
n		 17
n		 18
n		 19
n
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210
1 4 10 20 35 56 84 120 165 220 286 364 455 560 680 816 969 1140 1330 1540
1 5 15 35 70 126 210 330 495 715 1001 1365 1820 2380 3060 3876 4845 5985 7315 8855
1 6 21 56 126 252 462 792 1287 2002 3003 4368 6188 8568 11628 15504 20349 26334 33649 42504
1 7 28 84 210 462 924 1716 3003 5005 8008 12376 18564 27132 38760 54264 74613 100947 134596 177100
1 8 36 120 330 792 1716 3432 6435 11440 19448 31824 50388 77520 116280 170544 245157 346104 480700 657800
1 9 45 165 495 1287 3003 6435 12870 24310 43758 75582 125970 203490 319770 490314 735471 1081575 1562275 2220075
1 10 55 220 715 2002 5005 11440 24310 48620 92378 167960 293930 497420 817190 1307504 2042975 3124550 4686825 6906900
1 11 66 286 1001 3003 8008 19448 43758 92378 184756 352716 646646 1144066 1961256 3268760 5311735 8436285 13123110 20030010
1 12 78 364 1365 4368 12376 31824 75582 167960 352716 705432 1352078 2496144 4457400 7726160 13037895 21474180 34597290 54627300
1 13 91 455 1820 6188 18564 50388 125970 293030 646646 1352078 2704156 5200300 9657700 17383860 30421755 51895935 86493225 141120525
1 14 105 560 2380 8568 27132 77520 203490 497420 1144066 2496144 5200300 10400600 20058300 37442160 67863915 119759850 206253075 347373600
1 15 120 680 3060 11628 38760 116280 319770 817190 1961256 4457400 9657700 20058300 40116600 77558760 145422675 265182525 471435600 818809200
1 16 136 816 3876 15504 54264 170544 490314 1307504 3268760 7726160 17383860 37442160 77558760 155117520 300540195 565722720 1037158320 1855967520
1 17 153 969 4845 20349 74613 245157 735471 2042975 5311735 13037895 30421755 67863915 145422675 300540195 601080390 1166803110 2203961430 4059928950
1 18 171 1140 5985 26334 100947 346104 1081575 3124550 8436285 21474180 51895935 119759850 265182525 565722720 1166803110 2333606220 4537567650 8597496600
1 19 190 1330 7315 33649 134596 480700 1562275 4686825 13123110 34597290 86493225 206253075 471435600 1037158320 2203961430 4537567650 9075135300 17672631900
1 20 210 1540 8855 42504 177100 657800 2220075 6906900 20030010 54627300 141120525 347373600 818809200 1855967520 4059928950 8597496600 17672631900 35345263800
 
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