What is the term for a factorial type operation, but with summation instead of products? [duplicate]
The name for
$$ T_n= \sum_{k=1}^n k = 1+2+3+ \dotsb +(n-1)+n = \frac{n(n+1)}{2} = \frac{n^2+n}{2} = {n+1 \choose 2} $$
is the $n$th triangular number. This picture demonstrates the reasoning for the name:
$$T_1=1\qquad T_2=3\qquad T_3=6\qquad T_4=10\qquad T_5=15\qquad T_6=21$$
$\hskip1.7in$ 
Donald Knuth in The Art of Computer Programming calls the $n$-th triangular number the "termial function", and denotes it
$$n? = 1 + 2 + ... + n = \sum_{k=1}^n k $$
(Third edition, Volume 1, page 48).