This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334637 #41 Nov 05 2024 18:57:06
%S A334637 1,3,13,75,517,4443,43093,486315,6082117,81407163,1184034613,
%T A334637 19251200715,342825926437,6604284459483,136398242877973,
%U A334637 2984396941441515,68215762130020357,1627134074774283003,40749275946991321333,1079215210446044648715,30311064871950344936677,897713839789350372765723
%N A334637 Sum of different values of x_1*x_2*...*x_n where x_1=1 and x_i-x_{i-1} is 0 or 1.
%C A334637 Equals to: sum of different possible product of nesting levels in n pairs of parentheses.
%C A334637 For example, there are A000108(3)=5 ways to insert 3 pair of parentheses: ()()(), (())(), ()(()), (()()), ((())), the product of nesting levels are 1, 2, 2, 4, 6, and A001147(3)=1+2+2+4+6=15, but a(3)=1+2+4+6=13.
%H A334637 Using your Head is Permitted, <a href="https://www.brand.site.co.il/riddles/201311q.html">Parenthesis Values</a>, November 2013 riddle.
%e A334637 n=5: possible values are 1*1*1*1*1, 1*1*1*1*2, 1*1*1*2*2, 1*1*1*2*3, 1*1*2*2*2, 1*1*2*2*3, 1*1*2*3*3, 1*1*2*3*4, 1*2*2*2*2, 1*2*2*2*3, 1*2*2*3*3, 1*2*2*3*4, 1*2*3*3*3, 1*2*3*3*4, 1*2*3*4*4, 1*2*3*4*5, but since 1*1*2*3*4=1*2*2*2*3, the sum of other values is A000670(5)-1*1*2*3*4=517.
%o A334637 (Python)
%o A334637 k=[{(1, 1)}]
%o A334637 for i in range(20):
%o A334637 k.append(set([(i[0]*i[1], i[1]) for i in k[-1]])|set([(i[0]*(i[1]+1), i[1]+1) for i in k[-1]]))
%o A334637 [sum(set(j[0] for j in i)) for i in k]
%Y A334637 Cf. A334635 (number of different values), A000670 (sum if the values are not deduplicated), A001147 (sum of products of nesting levels in n pairs of parentheses if not deduplicated).
%K A334637 nonn
%O A334637 1,2
%A A334637 _Jack Zhang_, Sep 10 2020