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National and Regional Contests
China Contests
China Team Selection Test
2024 China Team Selection Test
13
Catalan identity
Catalan identity
Source: 2024 CTST P13
March 24, 2024
combinatorics
Problem Statement
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
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