MathDB

3

Part of 2013 Iran Team Selection Test

Problems(1)

Roots of a polynomial defined by a sequence are real

Source: Iran TST 2013: TST 1, Day 1, Problem 3

4/17/2013
For nonnegative integers mm and nn, define the sequence a(m,n)a(m,n) of real numbers as follows. Set a(0,0)=2a(0,0)=2 and for every natural number nn, set a(0,n)=1a(0,n)=1 and a(n,0)=2a(n,0)=2. Then for m,n≥1m,n\geq1, define a(m,n)=a(m−1,n)+a(m,n−1). a(m,n)=a(m-1,n)+a(m,n-1). Prove that for every natural number kk, all the roots of the polynomial Pk(x)=∑i=0ka(i,2k+1−2i)xiP_{k}(x)=\sum_{i=0}^{k}a(i,2k+1-2i)x^{i} are real.
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