MathDB

Let

m \leq n

be natural numbers. Starting with the product

t=m\cdot (m+1) \cdot (m+2) \cdot \cdots \cdot n

, let

T_{m, n}

be the sum of products that can be obtained from deleting from

t

pairs of consecutive integers (this includes

t

itself). In the case where all the numbers are deleted, we assume the number

1

.
For example,

T_{2, 7} = 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 + 2 \cdot 3 \cdot 4 \cdot 5 + 2 \cdot 3 \cdot 4 \cdot 7 + 2 \cdot 3 \cdot 6 \cdot 7 + 2 \cdot 5 \cdot 6 \cdot 7 + 4 \cdot 5 \cdot 6 \cdot 7 + 2 \cdot 3 + 2 \cdot 5 + 2 \cdot 7 + 4 \cdot 7 + 6 \cdot 7 + 1 = 5040 + 120 + 168 + 252 + 420 + 840 + 6 + 10 + 14 + 20 + 28 + 42 + 1 = 6961

.
Taking

T_{n+1, n} = 1

.
Show that

T_{m, n+1}=T_{m, k-1} \cdot T_{k+2, n+1} + T_{m, k} \cdot T_{k+1, n+1}

for all

1 \leq m \leq k \leq n

Problem Statement