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(a) Compute the sums of the squares of Rows 1-4 of Pascal's Triangle. That is, compute: Do these sums appear anywhere else in Pascal's Triangle? (b) Guess at an identity based on your observations from part (a). Your identity should be of the form (You have to figure out what "something" is.) Test your identity for using your results from part (a). (c) Prove your identity using a committee-forming argument. (d) Prove your identity using a block-walking argument. I only need help with part C!

 Jul 30, 2019
 #1
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by the way my equation is $C(n,0)^2$,$C(n,1)^2$,$C(n,2)^2$...$C(n,n)^2$=$C(2n,n)$

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 Jul 30, 2019
 #2
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and actually part D too, sorry!

 Jul 30, 2019
 #3
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This might help you. Here are the first 10 rows of Pascal's Triangle and some rules about the rows of the Triangle:
1   = R 0
1   1   = R 1
1   2   1   = R 2
1   3   3   1   = R 3
1   4   6   4   1   = R 4
1   5   10   10   5   1   = R 5
1   6   15   20   15   6   1   = R 6
1   7   21   35   35   21   7   1   = R 7
1   8   28   56   70   56   28   8   1   = R 8
1   9   36   84   126   126   84   36   9   1   = R 9
1   10   45   120   210   252   210   120   45   10   1   = R 10
The sum of the nth row is 2^n. 
The sum of rows 0 through n is 2^(n + 1) - 1.
The generating function of the nth row is (x + 1)^n. 

 Jul 30, 2019
 #4
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I'm sorry but this dosent help me. :)

 Jul 31, 2019
 #5
avatar+103674 
+1

1      sum of squares is 1         

1 1      sum of squares is 2            

1 2 1       sum of squares is 6           

1 3 3 1       sum of squares is 20         

1 4 6 4 1       sum of squares is 70        Edited, Thanks Heureka :))

 Aug 1, 2019
edited by Melody  Aug 2, 2019
 #6
avatar+23041 
+2

(a)

Compute the sums of the squares of Rows 1-4 of Pascal's Triangle.

 

Central binomial coefficients: \(\dbinom{2n}{n}\)

see:  http://oeis.org/search?q=1%2C2%2C6%2C20%2C70&sort=&language=english&go=Suche

 

\(\begin{array}{|rcll|} \hline \large{\sum \limits_{k=0}^{n} \dbinom{n}{k}^2 = \dbinom{2n}{n} }\\ \hline \end{array}\)

 

laugh

 Aug 2, 2019
edited by heureka  Aug 2, 2019
edited by heureka  Aug 2, 2019

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