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

rbzima
Messages
83
Reaction score
0
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
.
.
.
​
 
Physics news on Phys.org
Why have you posed this?

(nCk)2
 
mathman said:
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!
 
rbzima said:
Is there any particular pattern also with the sum of the rows as well?

See http://www.research.att.com/~njas/sequences/A000984" .
 
Last edited by a moderator:

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Row space, Column space, Null space & Left null space
Replies
8
Views
3K
I Dimension and solution to matrix-vector product
Replies
4
Views
3K
B How to interpret Pascal's Triangle for negative numbers?
Replies
3
Views
2K
B Is Each Stage of Pascal's Triangle Special?
Replies
11
Views
2K
Find the limit of the sum of the areas of all these triangles
Replies
7
Views
1K
Conjecture: Prime Divisibility & First Differences of Stirling & Eulerian Triangles
Replies
1
Views
3K
B Array Representation Of A General Tensor Question
Replies
2
Views
2K
Combinatorics: Counting quadrilaterals in triangle pattern
Replies
2
Views
2K
B Is it invalid to redefine the sgn function in this way in a proof?
Replies
15
Views
4K
MHB How can the convexity of a disk be proved using linear algebra?
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top