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 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. 3
 1, 4, 9, 19, 1, 33, 2, 59, 7, 93, 12, 150, 26, 226, 43, 1, 342, 76, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Column k starts in row k^2. It appears that column 1 gives A179862. LINKS EXAMPLE Triangle begins:     1;     4;     9;    19,  1;    33,  2;    59,  7;    93, 12;   150, 26;   226, 43, 1;   342, 76, 2; ... 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. . .                                     Right-angles   Right Part   Ferrers diagram         Part   diagram        angle                                       _ _ _ _ _ _ _   7    * * * * * * *             7   |  _ _ _ _ _ _|  14   6    * * * * * *               6   | |  _ _ _ _|     8   3    * * *                     3   | | | |           2   3    * * *                     3   | | |_|   2    * *                       2   | |_|   1    *                         1   | |   1    *                         1   | |   1    *                         1   |_| .        Figure 1.                      Figure 2. . For n = 8 the partitions of 8 and their respective right-angles diagrams are as follows: .     _       _ _       _ _ _       _ _ _ _       _ _ _ _ _   1| |8   2|  _|8   3|  _ _|8   4|  _ _ _|8   5|  _ _ _ _|8   1| |    1| |      1| |        1| |          1| |   1| |    1| |      1| |        1| |          1| |   1| |    1| |      1| |        1| |          1|_|   1| |    1| |      1| |        1|_|   1| |    1| |      1|_|   1| |    1|_|   1|_|     _ _ _ _ _ _       _ _ _ _ _ _ _       _ _ _ _ _ _ _ _   6|  _ _ _ _ _|8   7|  _ _ _ _ _ _|8   8|_ _ _ _ _ _ _ _|8   1| |              1|_|   1|_| .     _ _       _ _ _       _ _ _ _       _ _ _ _ _       _ _ _ _ _ _   2|  _|7   3|  _ _|7   4|  _ _ _|7   5|  _ _ _ _|7   6|  _ _ _ _ _|7   2| |_|1   2| |_|  1   2| |_|    1   2| |_|      1   2|_|_|        1   1| |      1| |        1| |          1|_|   1| |      1| |        1|_|   1| |      1|_|   1|_| .     _ _       _ _ _       _ _ _       _ _ _ _       _ _ _ _       _ _ _ _ _   2|  _|6   3|  _ _|6   3|  _ _|6   4|  _ _ _|6   4|  _ _ _|6   5|  _ _ _ _|6   2| | |2   2| | |  2   3| |_ _|2   2| | |    2   3| |_ _|  2   3|_|_ _|    2   2| |_|    2| |_|      1| |        2|_|_|        1|_|   1| |      1|_|        1|_|   1|_| .     _ _       _ _ _        _ _ _ _   2|  _|5   3|  _ _|5    4|  _ _ _|5   2| | |3   3| |  _|3    4|_|_ _ _|3   2| | |    2|_|_|   2|_|_| . 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. 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. CROSSREFS Row sums give A066186. Cf. A179862. Cf. also A000041, A330369, A330378, A330379. Sequence in context: A146303 A344999 A203205 * A147977 A045278 A184723 Adjacent sequences:  A330372 A330373 A330374 * A330376 A330377 A330378 KEYWORD nonn,tabf,more AUTHOR Omar E. Pol, Dec 21 2019 STATUS approved

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Last modified June 29 23:00 EDT 2022. Contains 354913 sequences. (Running on oeis4.)