A228527 Triangle read by rows: T(n,k) is the sum of all parts of size k of the n-th section of the set of compositions ( ordered partitions) of any integer >= n.
1, 1, 2, 3, 2, 3, 7, 6, 3, 4, 16, 14, 9, 4, 5, 36, 32, 21, 12, 5, 6, 80, 72, 48, 28, 15, 6, 7, 176, 160, 108, 64, 35, 18, 7, 8, 384, 352, 240, 144, 80, 42, 21, 8, 9, 832, 768, 528, 320, 180, 96, 49, 24, 9, 10, 1792, 1664, 1152, 704, 400, 216, 112, 56, 27, 10, 11
Offset: 1
Examples
Illustration (using the colexicograpical order of compositions A228525) of the four sections of the set of compositions of 4: . . 1 2 3 4 . _ _ _ _ . |_| _| | | | | | . |_ _| _ _| | | | . |_| | | | . |_ _ _| _ _ _| | . |_| | | . |_ _| | . |_| | . |_ _ _ _| . For n = 4 and k = 2, T(4,2) = 6 because there are 3 parts of size 2 in the last section of the set of compositions of 4, so T(4,2) = 3*2 = 6, see below: -------------------------------------------------------- . The last section Sum of . Composition of 4 of the set of parts of . compositions of 4 size k . -------------------- ------------------- . Diagram Diagram k = 1 2 3 4 . ------------------------------------------------------ . _ _ _ _ _ . 1+1+1+1 |_| | | | 1 | | 1 0 0 0 . 2+1+1 |_ _| | | 1 | | 1 0 0 0 . 1+2+1 |_| | | 1 | | 1 0 0 0 . 3+1 |_ _ _| | 1 _ _ _| | 1 0 0 0 . 1+1+2 |_| | | 1+1+2 |_| | | 2 2 0 0 . 2+2 |_ _| | 2+2 |_ _| | 0 4 0 0 . 1+3 |_| | 1+3 |_| | 1 0 3 0 . 4 |_ _ _ _| 4 |_ _ _ _| 0 0 0 4 . --------- . Column sums give row 4: 7,6,3,4 . Triangle begins: 1; 1, 2; 3, 2, 3; 7, 6, 3, 4; 16, 14, 9, 4, 5; 36, 32, 21, 12, 5, 6; 80, 72, 48, 28, 15, 6, 7; 176, 160, 108, 64, 35, 18, 7, 8; 384, 352, 240, 144, 80, 42, 21, 8, 9; 832, 768, 528, 320, 180, 96, 49, 24, 9, 10; 1792, 1664, 1152, 704, 400, 216, 112, 56, 27, 10, 11; ...
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