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.

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

Offset: 1

Author

Omar E. Pol, Sep 01 2013

Keywords

• nonn
• tabl

Comments

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.

Examples

			Illustration (using the colexicograpical order of compositions A228525) of the four sections of the set of compositions of 4:
.
.            1      2        3          4
.            _      _        _          _
.           |_|   _| |      | |        | |
.                |_ _|   _ _| |        | |
.                       |_|   |        | |
.                       |_ _ _|   _ _ _| |
.                                |_| |   |
.                                |_ _|   |
.                                |_|     |
.                                |_ _ _ _|
.
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
.
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;
...
		

Crossrefs

Column 1 is A045891. Row sums give A001792.

Cf. A011782, A135010, A207383, A221876, A228350, A228366, A228370, A228524, A228526.

Formula

T(n,k) = k*A045891(n-k) = k*A228524(n,k), n>=1, 1<=k<=n.