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%I #23 Sep 22 2013 16:22:44
%S 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,
%T 176,160,108,64,35,18,7,8,384,352,240,144,80,42,21,8,9,832,768,528,
%U 320,180,96,49,24,9,10,1792,1664,1152,704,400,216,112,56,27,10,11
%N 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.
%C In other words, T(n,k) is the sum of all parts of size k of the last section of the set of compositions (ordered partitions) of n.
%C For the definition of "section of the set of compositions" see A228524.
%C The equivalent sequence for partitions is A207383.
%F T(n,k) = k*A045891(n-k) = k*A228524(n,k), n>=1, 1<=k<=n.
%e Illustration (using the colexicograpical order of compositions A228525) of the four sections of the set of compositions of 4:
%e .
%e . 1 2 3 4
%e . _ _ _ _
%e . |_| _| | | | | |
%e . |_ _| _ _| | | |
%e . |_| | | |
%e . |_ _ _| _ _ _| |
%e . |_| | |
%e . |_ _| |
%e . |_| |
%e . |_ _ _ _|
%e .
%e 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:
%e --------------------------------------------------------
%e . The last section Sum of
%e . Composition of 4 of the set of parts of
%e . compositions of 4 size k
%e . -------------------- -------------------
%e . Diagram Diagram k = 1 2 3 4
%e . ------------------------------------------------------
%e . _ _ _ _ _
%e . 1+1+1+1 |_| | | | 1 | | 1 0 0 0
%e . 2+1+1 |_ _| | | 1 | | 1 0 0 0
%e . 1+2+1 |_| | | 1 | | 1 0 0 0
%e . 3+1 |_ _ _| | 1 _ _ _| | 1 0 0 0
%e . 1+1+2 |_| | | 1+1+2 |_| | | 2 2 0 0
%e . 2+2 |_ _| | 2+2 |_ _| | 0 4 0 0
%e . 1+3 |_| | 1+3 |_| | 1 0 3 0
%e . 4 |_ _ _ _| 4 |_ _ _ _| 0 0 0 4
%e . ---------
%e . Column sums give row 4: 7,6,3,4
%e .
%e Triangle begins:
%e 1;
%e 1, 2;
%e 3, 2, 3;
%e 7, 6, 3, 4;
%e 16, 14, 9, 4, 5;
%e 36, 32, 21, 12, 5, 6;
%e 80, 72, 48, 28, 15, 6, 7;
%e 176, 160, 108, 64, 35, 18, 7, 8;
%e 384, 352, 240, 144, 80, 42, 21, 8, 9;
%e 832, 768, 528, 320, 180, 96, 49, 24, 9, 10;
%e 1792, 1664, 1152, 704, 400, 216, 112, 56, 27, 10, 11;
%e ...
%Y Column 1 is A045891. Row sums give A001792.
%Y Cf. A011782, A135010, A207383, A221876, A228350, A228366, A228370, A228524, A228526.
%K nonn,tabl
%O 1,3
%A _Omar E. Pol_, Sep 01 2013
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