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Putnam
1965 Putnam
A2
A2
Part of
1965 Putnam
Problems
(1)
Putnam 1965 A2
Source:
9/27/2020
Show that, for any positive integer
n
n
n
,
∑
r
=
0
[
(
n
−
1
)
/
2
]
{
n
−
2
r
n
(
n
r
)
}
2
=
1
n
(
2
n
−
2
n
−
1
)
,
\sum_{r=0}^{[(n-1)/2]}\left\{\frac{n-2r}n\binom nr\right\}^2 = \frac 1n\binom{2n-2}{n-1},
r
=
0
∑
[(
n
−
1
)
/2
]
{
n
n
−
2
r
(
r
n
)
}
2
=
n
1
(
n
−
1
2
n
−
2
)
,
where
[
x
]
[x]
[
x
]
means the greatest integer not exceeding
x
x
x
, and
(
n
r
)
\textstyle\binom nr
(
r
n
)
is the binomial coefficient "
n
n
n
choose
r
r
r
", with the convention
(
n
0
)
=
1
\textstyle\binom n0 = 1
(
0
n
)
=
1
.
Putnam
binomial coefficients