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%I #27 Sep 22 2013 04:04:20
%S 1,2,2,5,4,3,12,10,6,4,28,24,15,8,5,64,56,36,20,10,6,144,128,84,48,25,
%T 12,7,320,288,192,112,60,30,14,8,704,640,432,256,140,72,35,16,9,1536,
%U 1408,960,576,320,168,84,40,18,10,3328,3072,2112,1280,720
%N Triangle read by rows: T(n,k) = sum of all parts of size k in all compositions (ordered partitions) of n.
%C The equivalent sequence for partitions is A138785, see the first comment there.
%F T(n,k) = k*A045623(n-k) = k*A221876(n,k), n >=1, 1<=k<=n.
%e T(4,2) = 10 because there are 5 parts of size 2 in all compositions of 4, T(4,2) = 5*2 = 10 (see below):
%e ---------------------------------------------------------
%e . Compositions Parts Sum of parts
%e . of 4 Diagram of size 2 of size 2
%e ---------------------------------------------------------
%e . _ _ _ _
%e . 1+1+1+1 |_| | | | 0 0
%e . 2+1+1 |_ _| | | 1 2
%e . 1+2+1 |_| | | 1 2
%e . 3+1 |_ _ _| | 0 0
%e . 1+1+2 |_| | | 1 2
%e . 2+2 |_ _| | 2 4
%e . 1+3 |_| | 0 0
%e . 4 |_ _ _ _| 0 0
%e . ----- ------
%e . Total 5 10
%e .
%e Triangle begins:
%e 1;
%e 2, 2;
%e 5, 4, 3;
%e 12, 10, 6, 4;
%e 28, 24, 15, 8, 5;
%e 64, 56, 36, 20, 10, 6;
%e 144, 128, 84, 48, 25, 12, 7;
%e 320, 288, 192, 112, 60, 30, 14, 8;
%e 704, 640, 432, 256, 140, 72, 35, 16, 9;
%e 1536, 1408, 960, 576, 320, 168, 84, 40, 18, 10;
%e 3328, 3072, 2112, 1280, 720, 384, 196, 96, 45, 20, 11;
%e ...
%Y Column k is k*A045623. Row sums give A001787, n >= 1. Right border gives A000027.
%Y Cf. A001792, A011782, A138785, A221876, A228525, A228527.
%K nonn,tabl
%O 1,2
%A _Omar E. Pol_, Aug 28 2013
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