Not sure what they represent, but I figured out how to generate them...
take a rectangular region of Pascal's Triangle...
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
5 10 10 5 1
15 20 15 6 1
35 35 21 7 1
70 56 28 8
126 84 36
210 120
330
and multiply each term by the corresponding term in the
flipped around version of the rectangle...
330
120 210
36 84 126
8 28 56 70
1 7 21 35 35
1 6 15 20 15
1 5 10 10 5
1 4 6 4 1
1 3 3 1
1 2 1
1 1
1
The terms in the array behave like the terms
in a "weighted" Pascal's triangle, with the weights
shown in parentheses below...
330
/ \
(4/11)(7/11)
/ \
120 210
/ \ / \
(3/10)(7/10)(4/10)(6/10)
/ \ / \
36 168 126
/ \ / \ / \
(2/9)(7/9) (3/9) (6/9) (4/9) (5/9)
/ \ / \ / \
8 84 168 70
/ \ / \ / \ / \
(1/8)(7/8) (2/8)(6/8) (3/8) (5/8) (4/8)(4/8)
/ \ / \ / \ / \
1 28 126 140 35
\ / \ / \ / \ / \
(7/7)(1/7) (6/7)(2/7) (5/7) (3/7) (4/7)(5/7)(2/7)
\ / \ / \ / \ / \
5 60 150 100 15
\ / \ / \ / \ / \
(6/6) (1/6)(5/6) (2/6) (4/6) (3/6)(3/6)(4/6)(2/6)
\ / \ / \ / \ / \
15 100 150 60 5
\ / \ / \ / \ / \
(5/5)(1/5) (4/5) (2/5) (3/5)(3/5)(2/5)(4/5)(1/5)
\ / \ / \ / \ / \
35 140 126 28 1
\ / \ / \ / \ /
(4/4) (1/4) (3/4) (2/4)(2/4)(3/4)(1/4)(4/4)
\ / \ / \ / \ /
70 168 84 8
\ / \ / \ /
(3/3) (1/3) (2/3)(2/3)(1/3)(3/3)
\ / \ / \ /
126 168 36
\ / \ /
(2/2) (1/2)(1/2)(2/2)
\ / \ /
210 120
\ /
(1/1)(1/1)
\ /
330
210
/ \
(4/10)(6/10)
/ \
84 126
/ \ / \
(3/9)(6/9) (4/9) (5/9)
/ \ / \
28 112 70
/ \ / \ / \
(2/8)(6/8)(3/8) (5/8) (4/8)(4/8)
/ \ / \ / \
7 63 105 35
/ \ / \ / \ / \
(1/7)(6/7)(2/7)(5/7) (3/7) (4/7)(5/7)(2/7)
/ \ / \ / \ / \
1 24 90 80 15
\ / \ / \ / \ / \
(6/6)(1/6)(5/6) (2/6) (4/6) (3/6)(3/6)(4/6)(2/6)
\ / \ / \ / \ / \
5 50 100 50 5
\ / \ / \ / \ / \
(5/5)(1/5) (4/5) (2/5) (3/5)(3/5)(2/5)(4/5)(1/5)
\ / \ / \ / \ / \
15 80 90 24 1
\ / \ / \ / \ /
(4/4)(1/4) (3/4) (2/4)(2/4)(3/4)(1/4)(4/4)
\ / \ / \ / \ /
35 105 63 7
\ / \ / \ /
(3/3) (1/3) (2/3)(2/3)(1/3)(3/3)
\ / \ / \ /
70 112 28
\ / \ /
(2/2) (1/2)(1/2)(2/2)
\ / \ /
126 84
\ /
(1/1)(1/1)
\ /
210
Note that the weights beneath each term sum to 1.
Andrew
On Sat, Apr 5, 2014 at 4:25 PM, Brendan McKay <Brendan.McKay at anu.edu.au>wrote:
> I have some finite arrays of numbers that I am trying to identify.
> They arise in the theory of graph reconstruction. There is
> a whole lot of them but I can't give a general description.
>
> One example:
>
> 330
> 120 210
> 36 168 126
> 8 84 168 70
> 1 28 126 140 35
> 5 60 150 100 15
> 15 100 150 60 5
> 35 140 126 28 1
> 70 168 84 8
> 126 168 36
> 210 120
> 330
>
> Another:
>
> 210
> 84 126
> 28 112 70
> 7 63 105 35
> 1 24 90 80 15
> 5 50 100 50 5
> 15 80 90 24 1
> 35 105 63 7
> 70 112 28
> 126 84
> 210
>
> Does this ring a bell for anyone? Can anyone guess what they are?
>
> Thanks, Brendan.
>
>
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