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 A330375 #25 Nov 20 2020 21:22:32 %S A330375 1,4,9,19,1,33,2,59,7,93,12,150,26,226,43,1,342,76,2 %N A330375 Irregular triangle read by rows: T(n,k) (n>=1) is the sum of the lengths of all k-th right angles in all partitions of n. %C A330375 Column k starts in row k^2. %C A330375 It appears that column 1 gives A179862. %e A330375 Triangle begins: %e A330375 1; %e A330375 4; %e A330375 9; %e A330375 19, 1; %e A330375 33, 2; %e A330375 59, 7; %e A330375 93, 12; %e A330375 150, 26; %e A330375 226, 43, 1; %e A330375 342, 76, 2; %e A330375 ... %e A330375 Figure 1 shows the Ferrers diagram of the partition of 24: [7, 6, 3, 3, 2, 1, 1, 1]. Figure 2 shows the right-angles diagram of the same partition. Note that in this last diagram we can see the size of the three right angles as follows: the first right angle has size 14 because it contains 14 square cells, the second right angle has size 8 and the third right angle has size 2. %e A330375 . %e A330375 . Right-angles Right %e A330375 Part Ferrers diagram Part diagram angle %e A330375 _ _ _ _ _ _ _ %e A330375 7 * * * * * * * 7 | _ _ _ _ _ _| 14 %e A330375 6 * * * * * * 6 | | _ _ _ _| 8 %e A330375 3 * * * 3 | | | | 2 %e A330375 3 * * * 3 | | |_| %e A330375 2 * * 2 | |_| %e A330375 1 * 1 | | %e A330375 1 * 1 | | %e A330375 1 * 1 |_| %e A330375 . %e A330375 Figure 1. Figure 2. %e A330375 . %e A330375 For n = 8 the partitions of 8 and their respective right-angles diagrams are as follows: %e A330375 . %e A330375 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A330375 1| |8 2| _|8 3| _ _|8 4| _ _ _|8 5| _ _ _ _|8 %e A330375 1| | 1| | 1| | 1| | 1| | %e A330375 1| | 1| | 1| | 1| | 1| | %e A330375 1| | 1| | 1| | 1| | 1|_| %e A330375 1| | 1| | 1| | 1|_| %e A330375 1| | 1| | 1|_| %e A330375 1| | 1|_| %e A330375 1|_| %e A330375 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A330375 6| _ _ _ _ _|8 7| _ _ _ _ _ _|8 8|_ _ _ _ _ _ _ _|8 %e A330375 1| | 1|_| %e A330375 1|_| %e A330375 . %e A330375 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A330375 2| _|7 3| _ _|7 4| _ _ _|7 5| _ _ _ _|7 6| _ _ _ _ _|7 %e A330375 2| |_|1 2| |_| 1 2| |_| 1 2| |_| 1 2|_|_| 1 %e A330375 1| | 1| | 1| | 1|_| %e A330375 1| | 1| | 1|_| %e A330375 1| | 1|_| %e A330375 1|_| %e A330375 . %e A330375 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A330375 2| _|6 3| _ _|6 3| _ _|6 4| _ _ _|6 4| _ _ _|6 5| _ _ _ _|6 %e A330375 2| | |2 2| | | 2 3| |_ _|2 2| | | 2 3| |_ _| 2 3|_|_ _| 2 %e A330375 2| |_| 2| |_| 1| | 2|_|_| 1|_| %e A330375 1| | 1|_| 1|_| %e A330375 1|_| %e A330375 . %e A330375 _ _ _ _ _ _ _ _ _ %e A330375 2| _|5 3| _ _|5 4| _ _ _|5 %e A330375 2| | |3 3| | _|3 4|_|_ _ _|3 %e A330375 2| | | 2|_|_| %e A330375 2|_|_| %e A330375 . %e A330375 The sum of the lengths of the first right angles of all partitions of 8 is 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 7 + 7 + 7 + 7 + 7 + 6 + 6 + 6 + 6 + 6 + 6 + 5 + 5 + 5 = 150, so T(8,1) = 150. %e A330375 The sum of the second right angles of all partitions of 8 is 1 + 1 + 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 = 26, so T(8,2) = 26. %Y A330375 Row sums give A066186. %Y A330375 Cf. A179862. %Y A330375 Cf. also A000041, A330369, A330378, A330379. %K A330375 nonn,tabf,more %O A330375 1,2 %A A330375 _Omar E. Pol_, Dec 21 2019