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IMAR Test
2023 IMAR Test
P4
P4
Part of
2023 IMAR Test
Problems
(1)
Polynomial with squared binomial coefficients
Source: IMAR Test 2023 P4
12/16/2023
Let
n
n{}
n
be a non-negative integer and consider the standard power expansion of the following polynomial
∑
k
=
0
n
(
n
k
)
2
(
X
+
1
)
2
k
(
X
−
1
)
2
(
n
−
k
)
=
∑
k
=
0
2
n
a
k
X
k
.
\sum_{k=0}^n\binom{n}{k}^2(X+1)^{2k}(X-1)^{2(n-k)}=\sum_{k=0}^{2n}a_kX^k.
k
=
0
∑
n
(
k
n
)
2
(
X
+
1
)
2
k
(
X
−
1
)
2
(
n
−
k
)
=
k
=
0
∑
2
n
a
k
X
k
.
The coefficients
a
2
k
+
1
a_{2k+1}
a
2
k
+
1
all vanish since the polynomial is invariant under the change
X
↦
−
X
.
X\mapsto -X.
X
↦
−
X
.
Prove that the coefficients
a
2
k
a_{2k}
a
2
k
are all positive.
polynomial
algebra