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Vojtěch Jarník IMC
2013 VJIMC
Problem 4
a binomial sum in two variables
a binomial sum in two variables
Source: VJIMC 2013 1.4
May 31, 2021
Summation
Binomials
algebra
Problem Statement
Let
n
n
n
and
k
k
k
be positive integers. Evaluate the following sum
∑
j
=
0
k
(
k
j
)
2
(
n
+
2
k
−
j
2
k
)
\sum_{j=0}^k\binom kj^2\binom{n+2k-j}{2k}
j
=
0
∑
k
(
j
k
)
2
(
2
k
n
+
2
k
−
j
)
where
(
n
k
)
=
n
!
k
!
(
n
−
k
)
!
\binom nk=\frac{n!}{k!(n-k)!}
(
k
n
)
=
k
!
(
n
−
k
)!
n
!
.
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