Question 142
Cryptogram of a positive integer $n$ is an $n$-tuple $a = (a_1, a_2, \dots, a_n)$ of non-negative integers such that
$$
a_1 + 2a_2 + \dots + n a_n = n.
$$
Let $\mathcal{K}_n$ be the set of all cryptograms of the number $n$. For $a \in \mathcal{K}_n$ let $J(a)$ denote the number of occurrences of the number 1 in the cryptogram $a$. Prove that
$$
\sum_{a \in \mathcal{K}_n} J(a) = \sum_{a \in \mathcal{K}_{n+1}} a_2.
$$
$$
a_1 + 2a_2 + \dots + n a_n = n.
$$
Let $\mathcal{K}_n$ be the set of all cryptograms of the number $n$. For $a \in \mathcal{K}_n$ let $J(a)$ denote the number of occurrences of the number 1 in the cryptogram $a$. Prove that
$$
\sum_{a \in \mathcal{K}_n} J(a) = \sum_{a \in \mathcal{K}_{n+1}} a_2.
$$