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 A225597 Triangle read by rows: T(n,k) = total number of parts of all regions of the set of partitions of n whose largest part is k. 1
 1, 1, 2, 1, 2, 3, 1, 3, 3, 5, 1, 3, 4, 5, 7, 1, 4, 5, 7, 7, 11, 1, 4, 6, 8, 9, 11, 15, 1, 5, 7, 11, 10, 15, 15, 22, 1, 5, 9, 12, 13, 17, 19, 22, 30, 1, 6, 10, 16, 15, 22, 21, 29, 30, 42, 1, 6, 12, 18, 19, 25, 26, 32, 38, 42, 56, 1, 7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For the definition of "region" see A206437. T(n,k) is also the number of parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section). LINKS EXAMPLE For n = 5 and k = 3 the set of partitions of 5 contains two regions whose largest part is 3, they are third region which contains three parts [3, 1, 1] and the sixth region which contains only one part [3]. Therefore the total number of parts is 3 + 1 = 4, so T(5,3) = 4. . .    Diagram    Illustration of parts ending in column k: .    for n=5      k=1   k=2     k=3       k=4        k=5 .   _ _ _ _ _                                  _ _ _ _ _ .  |_ _ _    |                _ _ _           |_ _ _ _ _| .  |_ _ _|_  |               |_ _ _|  _ _ _ _       |_ _| .  |_ _    | |          _ _          |_ _ _ _|        |_| .  |_ _|_  | |         |_ _|  _ _ _      |_ _|        |_| .  |_ _  | | |          _ _  |_ _ _|       |_|        |_| .  |_  | | | |      _  |_ _|     |_|       |_|        |_| .  |_|_|_|_|_|     |_|   |_|     |_|       |_|        |_| . k = 1 2 3 4 5 . The 5th row lists:  1     3       4         5          7 . Triangle begins: 1; 1,  2; 1,  2,  3; 1,  3,  3,  5; 1,  3,  4,  5,  7; 1,  4,  5,  7,  7, 11; 1,  4,  6,  8,  9, 11, 15; 1,  5,  7, 11, 10, 15, 15, 22; 1,  5,  9, 12, 13, 17, 19, 22, 30; 1,  6, 10, 16, 15, 22, 21, 29, 30, 42; 1,  6, 12, 18, 19, 25, 26, 32, 38, 42, 56; 1,  7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77; CROSSREFS Column 1 is A000012. Column 2 are the numbers => 2 of A008619. Row sums give A006128, n>=1. Right border gives A000041, n>=1. Second right border gives A000041, n>=1. Cf. A006128, A133041, A135010, A138137, A139582, A141285, A182377, A186114, A186412, A187219, A193870, A194446,  A206437, A207779, A211978, A220517, A225598, A225600, A225610. Sequence in context: A196935 A226314 A036995 * A116908 A265146 A325537 Adjacent sequences:  A225594 A225595 A225596 * A225598 A225599 A225600 KEYWORD nonn,tabl AUTHOR Omar E. Pol, Aug 02 2013 STATUS approved

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Last modified June 17 22:17 EDT 2019. Contains 324200 sequences. (Running on oeis4.)