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 A330379 #31 Jan 22 2022 18:22:03 %S A330379 1,0,4,0,0,9,1,0,3,16,2,0,0,8,25,3,4,0,8,15,36,4,8,0,0,20,24,49,5,12, %T A330379 9,0,15,36,35,64,7,16,21,0,5,36,56,48,81,9,20,33,16,0,36,63,80,63,100, %U A330379 13,24,45,40,0,12,77,96,108,80,121 %N A330379 Triangle read by rows: T(n,k) (1 <= k <= n) is the sum of the sizes of all right angles of size k of all partitions of n. %C A330379 Observation: at least the first 11 terms of column 1 coincide with A188674 (using the same indices). %D A330379 G. E. Andrews, Theory of Partitions, Cambridge University Press, 1984, page 143. %F A330379 T(n,k) = k*A330369(n,k). %e A330379 Triangle begins: %e A330379 1; %e A330379 0, 4; %e A330379 0, 0, 9; %e A330379 1, 0, 3, 16; %e A330379 2, 0, 0, 8, 25; %e A330379 3, 4, 0, 8, 15, 36; %e A330379 4, 8, 0, 0, 20, 24, 49; %e A330379 5, 12, 9, 0, 15, 36, 35, 64; %e A330379 7, 16, 21, 0, 5, 36, 56, 48, 81; %e A330379 9, 20, 33, 16, 0, 36, 63, 80, 63, 100; %e A330379 13, 24, 45, 40, 0, 12, 77, 96, 108, 80, 121; %e A330379 ... %e A330379 Below the figure 1 shows the Ferrers diagram of the partition of 24: [7, 6, 3, 3, 2, 1, 1, 1]. The 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 A330379 . %e A330379 . Right-angles Right %e A330379 Part Ferrers diagram Part diagram angle %e A330379 _ _ _ _ _ _ _ %e A330379 7 * * * * * * * 7 | _ _ _ _ _ _| 14 %e A330379 6 * * * * * * 6 | | _ _ _ _| 8 %e A330379 3 * * * 3 | | | | 2 %e A330379 3 * * * 3 | | |_| %e A330379 2 * * 2 | |_| %e A330379 1 * 1 | | %e A330379 1 * 1 | | %e A330379 1 * 1 |_| %e A330379 . %e A330379 Figure 1. Figure 2. %e A330379 . %e A330379 For n = 8 the partitions of 8 and their respective right-angles diagrams look as shown below: %e A330379 . %e A330379 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A330379 1| |8 2| _|8 3| _ _|8 4| _ _ _|8 5| _ _ _ _|8 %e A330379 1| | 1| | 1| | 1| | 1| | %e A330379 1| | 1| | 1| | 1| | 1| | %e A330379 1| | 1| | 1| | 1| | 1|_| %e A330379 1| | 1| | 1| | 1|_| %e A330379 1| | 1| | 1|_| %e A330379 1| | 1|_| %e A330379 1|_| %e A330379 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A330379 6| _ _ _ _ _|8 7| _ _ _ _ _ _|8 8|_ _ _ _ _ _ _ _|8 %e A330379 1| | 1|_| %e A330379 1|_| %e A330379 . %e A330379 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A330379 2| _|7 3| _ _|7 4| _ _ _|7 5| _ _ _ _|7 6| _ _ _ _ _|7 %e A330379 2| |_|1 2| |_| 1 2| |_| 1 2| |_| 1 2|_|_| 1 %e A330379 1| | 1| | 1| | 1|_| %e A330379 1| | 1| | 1|_| %e A330379 1| | 1|_| %e A330379 1|_| %e A330379 . %e A330379 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A330379 2| _|6 3| _ _|6 3| _ _|6 4| _ _ _|6 4| _ _ _|6 5| _ _ _ _|6 %e A330379 2| | |2 2| | | 2 3| |_ _|2 2| | | 2 3| |_ _| 2 3|_|_ _| 2 %e A330379 2| |_| 2| |_| 1| | 2|_|_| 1|_| %e A330379 1| | 1|_| 1|_| %e A330379 1|_| %e A330379 . %e A330379 _ _ _ _ _ _ _ _ _ %e A330379 2| _|5 3| _ _|5 4| _ _ _|5 %e A330379 2| | |3 3| | _|3 4|_|_ _ _|3 %e A330379 2| | | 2|_|_| %e A330379 2|_|_| %e A330379 . %e A330379 There are 5 right angles of size 1, so T(8,1) = 5*1 = 5. %e A330379 There are 6 right angles of size 2, so T(8,2) = 6*2 = 12. %e A330379 There are 3 right angles of size 3, so T(8,3) = 3*3 = 9. %e A330379 There are no right angle of size 4, so T(8,4) = 0*4 = 0. %e A330379 There are 3 right angles of size 5, so T(8,5) = 3*5 = 15. %e A330379 There are 6 right angles of size 6, so T(8,6) = 6*6 = 36. %e A330379 There are 5 right angles of size 7, so T(8,7) = 5*7 = 35. %e A330379 There are 8 right angles of size 8, so T(8,8) = 8*8 = 64. %e A330379 Hence the 8th row of triangle is [5, 12, 9, 0, 15, 36, 35, 64]. %e A330379 The row sum gives A066186(8) = 8*A000041(8) = 8*22 = 176. %Y A330379 Row sums give A066186, n >= 1. %Y A330379 Row sums of the terms that are after last zero give A179862. %Y A330379 Cf. A188674. %Y A330379 Cf. A000041, A000290, A330369, A330375, A330378. %K A330379 nonn,tabl,more %O A330379 1,3 %A A330379 _Omar E. Pol_, Dec 31 2019