MathDB
Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2024 China Team Selection Test
13
13
Part of
2024 China Team Selection Test
Problems
(1)
Catalan identity
Source: 2024 CTST P13
3/24/2024
For a natural number
n
n
n
, let
C
n
=
1
n
+
1
(
2
n
n
)
=
(
2
n
)
!
n
!
(
n
+
1
)
!
C_n=\frac{1}{n+1}\binom{2n}{n}=\frac{(2n)!}{n!(n+1)!}
C
n
=
n
+
1
1
(
n
2
n
)
=
n
!
(
n
+
1
)!
(
2
n
)!
be the
n
n
n
-th Catalan number. Prove that for any natural number
m
m
m
,
∑
i
+
j
+
k
=
m
C
i
+
j
C
j
+
k
C
k
+
i
=
3
2
m
+
3
C
2
m
+
1
.
\sum_{i+j+k=m} C_{i+j}C_{j+k}C_{k+i}=\frac{3}{2m+3}C_{2m+1}.
i
+
j
+
k
=
m
∑
C
i
+
j
C
j
+
k
C
k
+
i
=
2
m
+
3
3
C
2
m
+
1
.
Proposed by Bin Wang
combinatorics