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A228526 Triangle read by rows: T(n,k) = sum of all parts of size k in all compositions (ordered partitions) of n.

%I A228526 #27 Sep 22 2013 04:04:20
%S A228526 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 A228526 12,7,320,288,192,112,60,30,14,8,704,640,432,256,140,72,35,16,9,1536,
%U A228526 1408,960,576,320,168,84,40,18,10,3328,3072,2112,1280,720
%N A228526 Triangle read by rows: T(n,k) = sum of all parts of size k in all compositions (ordered partitions) of n.
%C A228526 The equivalent sequence for partitions is A138785, see the first comment there.
%F A228526 T(n,k) = k*A045623(n-k) = k*A221876(n,k), n >=1, 1<=k<=n.
%e A228526 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 A228526 ---------------------------------------------------------
%e A228526 . Compositions                   Parts      Sum of parts
%e A228526 .     of 4        Diagram      of size 2     of size 2
%e A228526 ---------------------------------------------------------
%e A228526 .                 _ _ _ _
%e A228526 .   1+1+1+1      |_| | | |         0             0
%e A228526 .     2+1+1      |_ _| | |         1             2
%e A228526 .     1+2+1      |_|   | |         1             2
%e A228526 .       3+1      |_ _ _| |         0             0
%e A228526 .     1+1+2      |_| |   |         1             2
%e A228526 .       2+2      |_ _|   |         2             4
%e A228526 .       1+3      |_|     |         0             0
%e A228526 .         4      |_ _ _ _|         0             0
%e A228526 .                                -----        ------
%e A228526 .                           Total  5            10
%e A228526 .
%e A228526 Triangle begins:
%e A228526 1;
%e A228526 2,       2;
%e A228526 5,       4,    3;
%e A228526 12,     10,    6,    4;
%e A228526 28,     24,   15,    8,   5;
%e A228526 64,     56,   36,   20,  10,   6;
%e A228526 144,   128,   84,   48,  25,  12,   7;
%e A228526 320,   288,  192,  112,  60,  30,  14,  8;
%e A228526 704,   640,  432,  256, 140,  72,  35, 16,  9;
%e A228526 1536, 1408,  960,  576, 320, 168,  84, 40, 18, 10;
%e A228526 3328, 3072, 2112, 1280, 720, 384, 196, 96, 45, 20, 11;
%e A228526 ...
%Y A228526 Column k is k*A045623. Row sums give A001787, n >= 1. Right border gives A000027.
%Y A228526 Cf. A001792, A011782, A138785, A221876, A228525, A228527.
%K A228526 nonn,tabl
%O A228526 1,2
%A A228526 _Omar E. Pol_, Aug 28 2013