Triangle

[Gottfried Helms]

Am 16.04.2008 10:31 schrieb Gottfried Helms:
> 
>  X1 =  // Stirling numbers 2'nd kind, row shifted
>  | 1   .    .    .    .    . |
>  | .   1    1    .    .    . |
>  | .   .    1    3    1    . |
>  | .   .    .    1    7    6 |
>  | .   .    .    .    1   15 |
>  | .   .    .    .    .    1 |
> 
> Then simply
> 
>      H3 = St2 * X1
> 
> In a matrix-multiplication-scheme
>                              X1=
>                             | 1   .    .    .    .    . |
>                             | .   1    1    .    .    . |
>                             | .   .    1    3    1    . |
>                             | .   .    .    1    7    6 |
>                             | .   .    .    .    1   15 |
>                             | .   .    .    .    .    1 |
>   St2=                       H3 =
>  | 1   .   .   .   .  . |   | 1   .    .    .    .    . |
>  | 1   1   .   .   .  . |   | 1   1    1    .    .    . |
>  | 1   3   1   .   .  . |   | 1   3    4    3    1    . |
>  | 1   7   6   1   .  . |   | 1   7   13   19   13    6 |
>  | 1  15  25  10   1  . |   | 1  15   40   85   96   75 |
>  | 1  31  90  65  15  1 |   | 1  31  121  335  560  616 |
> 
> It is nice, that X1 is a simple rowshift of St2.
> Now it would be good, if also H4 could be computed by a
> similar simple scheme:
> 
> H4 =
> | 1   .   .    .    .    .    .    .    .   .   .   .  . |
> | 1   1   1    1    .    .    .    .    .   .   .   .  . |
> | 1   3   4    6    4    3    1    .    .   .   .   .  . |
> | 1   7  13   26   31   31   25   13    6   1   .   .  . |
> | 1  15  40  100  171  220  255  215  156  85  35  10  1 |
> 
> I fiddled a bit but didn't see it yet ...
> 

Well, that was easy. And seems to give the most simple
recursive generation scheme.

Row-shifting H3 in the obvious way gives X4
X4=
|  1  .  .  .  .   .   .   .   .   .   .   .  . |
|  .  1  1  1  .   .   .   .   .   .   .   .  . |
|  .  .  1  3  4   3   1   .   .   .   .   .  . |
|  .  .  .  1  7  13  19  13   6   1   .   .  . |
|  .  .  .  .  1  15  40  85  96  75  35  10  1 |

and then again simply

  H4 = St2 * X4

H4 =
| 1   .   .    .    .    .    .    .    .   .   .   .  . |
| 1   1   1    1    .    .    .    .    .   .   .   .  . |
| 1   3   4    6    4    3    1    .    .   .   .   .  . |
| 1   7  13   26   31   31   25   13    6   1   .   .  . |
| 1  15  40  100  171  220  255  215  156  85  35  10  1 |

Many thanks for your input!

Gottfried Helms