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A228527
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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.
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2
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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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OFFSET
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1,3
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COMMENTS
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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.
For the definition of "section of the set of compositions" see A228524.
The equivalent sequence for partitions is A207383.
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LINKS
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FORMULA
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EXAMPLE
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Illustration (using the colexicograpical order of compositions A228525) of the four sections of the set of compositions of 4:
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. 1 2 3 4
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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
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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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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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