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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.

%I A228527 #23 Sep 22 2013 16:22:44
%S A228527 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 A228527 176,160,108,64,35,18,7,8,384,352,240,144,80,42,21,8,9,832,768,528,
%U A228527 320,180,96,49,24,9,10,1792,1664,1152,704,400,216,112,56,27,10,11
%N 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.
%C A228527 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 A228527 For the definition of "section of the set of compositions" see A228524.
%C A228527 The equivalent sequence for partitions is A207383.
%F A228527 T(n,k) = k*A045891(n-k) = k*A228524(n,k), n>=1, 1<=k<=n.
%e A228527 Illustration (using the colexicograpical order of compositions A228525) of the four sections of the set of compositions of 4:
%e A228527 .
%e A228527 .            1      2        3          4
%e A228527 .            _      _        _          _
%e A228527 .           |_|   _| |      | |        | |
%e A228527 .                |_ _|   _ _| |        | |
%e A228527 .                       |_|   |        | |
%e A228527 .                       |_ _ _|   _ _ _| |
%e A228527 .                                |_| |   |
%e A228527 .                                |_ _|   |
%e A228527 .                                |_|     |
%e A228527 .                                |_ _ _ _|
%e A228527 .
%e A228527 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 A228527 --------------------------------------------------------
%e A228527 .                         The last section      Sum of
%e A228527 .   Composition of 4        of the set of      parts of
%e A228527 .                         compositions of 4     size k
%e A228527 . --------------------   -------------------
%e A228527 .            Diagram             Diagram    k = 1 2 3 4
%e A228527 . ------------------------------------------------------
%e A228527 .            _ _ _ _                    _
%e A228527 .  1+1+1+1  |_| | | |         1        | |      1 0 0 0
%e A228527 .    2+1+1  |_ _| | |         1        | |      1 0 0 0
%e A228527 .    1+2+1  |_|   | |         1        | |      1 0 0 0
%e A228527 .      3+1  |_ _ _| |         1   _ _ _| |      1 0 0 0
%e A228527 .    1+1+2  |_| |   |     1+1+2  |_| |   |      2 2 0 0
%e A228527 .      2+2  |_ _|   |       2+2  |_ _|   |      0 4 0 0
%e A228527 .      1+3  |_|     |       1+3  |_|     |      1 0 3 0
%e A228527 .        4  |_ _ _ _|         4  |_ _ _ _|      0 0 0 4
%e A228527 .                                              ---------
%e A228527 .                      Column sums give row 4:  7,6,3,4
%e A228527 .
%e A228527 Triangle begins:
%e A228527 1;
%e A228527 1,       2;
%e A228527 3,       2,    3;
%e A228527 7,       6,    3,   4;
%e A228527 16,     14,    9,   4,   5;
%e A228527 36,     32,   21,  12,   5,   6;
%e A228527 80,     72,   48,  28,  15,   6,   7;
%e A228527 176,   160,  108,  64,  35,  18,   7,  8;
%e A228527 384,   352,  240, 144,  80,  42,  21,  8,  9;
%e A228527 832,   768,  528, 320, 180,  96,  49, 24,  9, 10;
%e A228527 1792, 1664, 1152, 704, 400, 216, 112, 56, 27, 10, 11;
%e A228527 ...
%Y A228527 Column 1 is A045891. Row sums give A001792.
%Y A228527 Cf. A011782, A135010, A207383, A221876, A228350, A228366, A228370, A228524, A228526.
%K A228527 nonn,tabl
%O A228527 1,3
%A A228527 _Omar E. Pol_, Sep 01 2013