Wednesday, 31 May 2017

summation - Evaluating sum of binomial coefficients



This sum popped out of one of my calculations, I know what it should evaluate to, but I have no idea how to prove it.
ri=0(n2i)(n2i1)
I know that 2i1 is negative for i=0, but for the purpose of this sum, we will say that (nx)=0 if n<0 or x<0. So this sum is basically summing the difference of consecutive even/odd binomial coefficient pairs. We can rewrite this sum as



2ri=0(1)i(ni).



I don't really know how to proceed from here, I couldn't find any information on evaluating sums of alternating series involving binomial coefficients.


Answer




This is a special case of the result
mk=0(1)k(nk)=(1)m(n1m)
(0mn1) which can be proved by induction on m.


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