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        riddles >> medium >> Pascal's triangle
        
(Message started by: Christine on Sep 9^th, 2013, 11:32am)

Title: Pascal's triangle
Post by Christine on Sep 9^th, 2013, 11:32am

A073617 a(n) = product of terms at +45 degrees slope with the horizontal.
1, 1, 1, 2, 3, 12, 30, 240, 1050, 16800, 132300, 4233600, 61122600, 3911846400, 104886381600, 13425456844800, 674943865596000, 172785629592576000, 16407885372638760000, 8400837310791045120000,
1515727634953623371280000

http://oeis.org/A073617

Taking sets of 4 consecutive terms, then
the product of the 1-st and 4-th divided by the product of the 2-nd and 3-rd

1, 1, 1, 2
(1*2)/(1*1) = 2

1, 1, 2, 3
(1*3)/(1*2) = 3/2 = 1.5

1, 2, 3, 12
(1*12)/(2*3) = 2

2, 3, 12, 30
(2*30)/(3*12) = 5/3 = 1.666..

3, 12, 30, 240
(3*240)/(12*30) = 2

when starting with an even indexed row the ratio is 2

Will the ratio always be 2?

with odd indexed row, the ratios are

1.5
1.666..
1.75
1.8
1.83333...
1.857142...
1.875
1.88888...
1.90

Will the ratio reach 2 ?

Title: Re: Pascal's triangle
Post by towr on Sep 12^th, 2013, 1:02am

on 09/09/13 at 11:32:59, Christine wrote:

when starting with an even indexed row the ratio is 2

WolframAlpha (http://www.wolframalpha.com/input/?i=%28prod+k%3D1..m+C%282m-k%2Ck%29%2FC%282m-k%2B1%2Ck%29%29%2F%28prod+k%3D1..%28m%2B1%29+C%282m-k%2B2%2Ck%29%2FC%282m-k%2B3%2Ck%29%29) confirms that that's the case. I haven't managed to do the calculation by hand though. And the odd case is little more complex than wolframalpha is willing to do without further simplification.