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%I #18 Mar 13 2015 22:58:17
%S 1,1,3,1,3,5,1,5,5,9,1,5,8,9,12,1,7,11,15,12,20,1,7,14,19,19,20,25,1,
%T 9,17,29,24,33,25,38,1,9,23,33,36,42,39,38,49,1,11,26,47,46,61,49,61,
%U 49,69,1,11,32,55,63,76,70,76,76,69,87,1,13,38,73,78,110,87,111,95,108,87,123
%N Triangle read by rows: T(n,k) = sum of all parts of all regions of the set of partitions of n whose largest part is k.
%C For the definition of region see A206437.
%C T(n,k) is also the sum of all parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section).
%e 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 sum of all parts is 3 + 1 + 1 + 3 = 8, so T(5,3) = 8.
%e .
%e . Diagram Illustration of parts ending in column k:
%e . for n=5 k=1 k=2 k=3 k=4 k=5
%e . _ _ _ _ _ _ _ _ _ _
%e . |_ _ _ | _ _ _ |_ _ _ _ _|
%e . |_ _ _|_ | |_ _ _| _ _ _ _ |_ _|
%e . |_ _ | | _ _ |_ _ _ _| |_|
%e . |_ _|_ | | |_ _| _ _ _ |_ _| |_|
%e . |_ _ | | | _ _ |_ _ _| |_| |_|
%e . |_ | | | | _ |_ _| |_| |_| |_|
%e . |_|_|_|_|_| |_| |_| |_| |_| |_|
%e .
%e k = 1 2 3 4 5
%e .
%e The 5th row lists: 1 5 8 9 12
%e .
%e Triangle begins:
%e 1;
%e 1, 3;
%e 1, 3, 5;
%e 1, 5, 5, 9;
%e 1, 5, 8, 9, 12;
%e 1, 7, 11, 15, 12, 20;
%e 1, 7, 14, 19, 19, 20, 25;
%e 1, 9, 17, 29, 24, 33, 25, 38;
%e 1, 9, 23, 33, 36, 42, 39, 38, 49;
%e 1, 11, 26, 47, 46, 61, 49, 61, 49, 69;
%e 1, 11, 32, 55, 63, 76, 70, 76, 76, 69, 87;
%e 1, 13, 38, 73, 78, 110, 87, 111, 95, 108, 87, 123;
%Y Column 1 is A000012. Column 2 are the numbers >= 3 of A109613. Row sums give A066186. Right border gives A046746. Second right border gives A046746.
%Y Cf. A000041, A066186, A135010, A141285, A186114, A186412, A187219, A194446, A206437, A207779, A211978, A225597, A225600, A225610.
%K nonn,tabf
%O 1,3
%A _Omar E. Pol_, Aug 02 2013
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