Suppose that I have a product of, say, $n=4$ integers starting with one and ending with four $1234=4!=24$. Now I construct all products of four positive integers $1,2,3$ and $4$ with repetition such that the difference between the integer on the $k$-th and $(k-1)$-st position is less or equal to one for $k=2,\dots,n$.
The rule looks contrived so here is the list of all possibilities for $n=4$ to get an idea: $$ 1234\\ 1233\\ 1232\\ 1231\\ 1223\\ 1222\\ 1221\\ 1212\\ 1211\\ 1123\\ 1122\\ 1121\\ 1112\\ 1111 $$
My questions: Does such a sequence of products have a name (for arbitrary $n$) and is there a formula for the sum of the products?
The list has 14 products which is one of the Catalan numbers (not a coincidence).
well this is the best i could find online.
https://books.google.com/books?id=5NvrH4w8WGsC&pg=PA428&lpg=PA428&dq=1234+1233+1232+1231+1223+1222+1221+1212+1211+1123+1122+1121+1112+1111&source=bl&ots=WMylvj0faU&sig=z9HjmCWEKl80qCZxcXx2-GaxluY&hl=en&sa=X&ei=DuQTVcu4McnlUsuJgcAC&ved=0CCQQ6AEwAQ#v=onepage&q=1234%201233%201232%201231%201223%201222%201221%201212%201211%201123%201122%201121%201112%201111&f=false
maybe you have to give it a name yourself :D