[seqfan] Connection between A000245 and A026016 (or A071725)?

[Ali Sada]

Hi Everyone,

Arrange the numbers between 1 and n in a raw of size 2n such that: 
a) 1 is always in the first spot;
b) k's position in the raw is <= 2k; and  
c) k always comes before k+1


For example, below are the possibilities we get when n = 3. (I put “*” instead of a space for more clarity.) 

1, 2, 3, *,* ,* 
1, 2, *, 3, *, *
1, 2, *, *, 3, * 
1, 2, *, *, *, 3
1, *, 2, 3, *, * 
1, *, 2, *, 3, *
1, *, 2, *, *, 3
1, *, *, 2, 3, * 
1, *, *, 2, *, 3

The possibilities I got were 1, 3, 9, 28, 90, 297, 1001, 3432, 11934 (A000245.)

Then, I filled the remaining spaces with the numbers between n+1 and 2n. 

Using n = 3 as an example we get: 

1, 2, 3, 4, 5, 6
1, 2, 4, 3, 5, 6
1, 2, 4, 5, 3, 6
1, 2, 4, 5, 6, 3
1, 4, 2, 3, 5, 6
1, 4, 2, 5, 3, 6
1, 4, 2, 5, 6, 3
1, 4, 5, 2, 3, 6
1, 4, 5, 2, 6, 3

For the numbers I tested (only 8,) the sum of the third column minus the sum of the second column was A026016(n-1).

In the example of n = 3 above, the sum of the third column is 31, and the sum of the second column is 28. 
31-28 = 3, which is A026016(2).

3 is also the result of the subtraction of the fourth column minus the third column, and the sixth column minus the fifth column. 

I would really appreciate it if you could tell me if this pattern will continue and why.

Best,

Ali