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 |