How Can We Represent and Sum the Rows of This Manipulated Pascal's Triangle?

[rbzima]

Here is the manipulation of Pascals Triangle. I'm trying to figure out a way to represent these rows in terms of a combination. Check it out...

1
1 1
1 4 1
1 9 9 1
1 16 36 16 1
1 25 100 100 25 1
.
.
.
​

 

[mathman]

Why have you posed this?

(nCk)²

 

[rbzima]

Why have you posed this?

(nCk)²

Is there any particular pattern also with the sum of the rows as well?

The reason is because it's part of a problem I'm working on regarding two people at opposite corners on a massive n by n board. On player can only move north and east, the other south and west. Basically, I need to find the expected value of the number of times they will meet, and I have a hunch this plays a part in. Thanks!

 

[dodo]

Is there any particular pattern also with the sum of the rows as well?

See http://www.research.att.com/~njas/sequences/A000984" .