[seqfan] Re: a number triangle I don't (yet) understand

israel at math.ubc.ca israel at math.ubc.ca
Wed Jul 30 17:22:30 CEST 2014


The row sums are the values at z=1. Since phi(1)=1, the k=1 factor is 0 
there. The first moment is the derivative evaluated at z=1. This is 0 for 
n>1 because phi(1)=1 and phi(2)=1.

Cheers,
Robert Israel

On Jul 30 2014, Wouter Meeussen wrote:

>( ref: http://math.stackexchange.com/questions/801399)
>
>prod(k=0..n; 1 - eulerphi(k) * z)  gives a polynome in z with coefficients
>
>      1        
>      1 -1       
>      1 -2 1      
>      1 -4 5 -2     
>      1 -6 13 -12 4    
>      1 -10 37 -64 52 -16   
>      1 -12 57 -138 180 -120 32  
>      1 -18 129 -480 1008 -1200 752 -192 
>
>
>Why are all row sums for n>0 exactly zero?
>Why are all the moments for n>1 zero?   
>      1        
>      1 -2       
>      1 -4 3      
>      1 -8 15 -8     
>      1 -12 39 -48 20    
>      1 -20 111 -256 260 -96   
>      1 -24 171 -552 900 -720 224  
>      1 -36 387 -1920 5040 -7200 5264 -1536 
>
>
>How many number triangles have that property?
>I remember having seen trangles with that property before.
>
>I can't shake the feeling of overlooking something obvious here,
>so, if you see what it is, please tell me. I'm blushing preemptively here.
>
>Wouter.
>
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