For 0≤r<2n, 2n+rCn 2n−rCn cannot exceed

For 0≤r<2n,(1+x)4n=(1+x)2n+r(1+x)2n−r=A0+A1x+A2x2+…+A2n+rx2n+rB0+B1x+B2x2+…+B2n−rx2n−r where Ak=2n+rCk(0≤k≤2n+r)and Bk=2n−rCk(0≤k≤2n−r)Coefficient of x2n on the RHSA2nB0+A2n−1B1+…+AnBn+An+1Bn−1+…+ArB2n−r         = coefficient of x2n on LHS=4nC2n.Thus, AnBn<4nC2n ⇒ 2n+rCn 2n−rCn<4nC2n