TITLE: Represent $1^5+2^4+3^3+4^2+5^1$ in sum notation
QUESTION [5 upvotes]: I imagine the sum notation for 
$$1^5+2^4+3^3+4^2+5^1$$
Would look something like 
$$ \sum x^y,\ x=1 \text{ to } 5,\  y=5 \text{ to } 1 $$
Is this correct or am I missing something?

REPLY [10 votes]: Observe that in each term, the sum of the base and the exponent is $6$.  Thus, if the base is $k$, the exponent is $6 - k$.  Since the base increases from $1$ to $5$, we obtain
$$\sum_{k = 1}^{5} k^{6 - k}$$

REPLY [3 votes]: Two ways I can think of:
$$\sum_{k=1}^5 k^{6-k}$$
$$\sum_{a,b\in\mathbb{N}_{\ge1}: a+b=6}a^b$$

REPLY [2 votes]: $\sum_{i=1}^5 i^{6-i}$ is equal to your sum