Identify the amount $x_n$ of sequences $(a_1,\cdots,a_n)$ of integers $\;a_i \geq 0,\;\; 1 \leq i \leq n\;\;$ so that $\;\sum_{i=1}^j a_i \geq j$ for $1 \leq j \leq n - 1\;$ and $\;\sum_{i=1}^n a_i = n. \quad x_0$ is set as 1.
Hint: Try to guess the by first identifying the values for small $n$. Additionally you can use OEIS.
To recap all this I am looking for sequences with $n$ elements where applies:
- All members are positive integers (including 0)
- The sum of all elements is n
- $\sum_{i=1}^j a_i \geq j$ for $1 \leq j \leq n - 1$
As all members are at least 0, the last one is equal to "$ a_1 \geq j\;$", isn't it?
For $n=2$ the sequences are (1,1) and (2,0). For $n=3$ it's (1,1,1),(3,0,0),(2,1,0),(2,0,1),(1,2,0).
Am I right? Do you have any hint how to find the answer?