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

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, 12, 7, 320, 288, 192, 112, 60, 30, 14, 8, 704, 640, 432, 256, 140, 72, 35, 16, 9, 1536, 1408, 960, 576, 320, 168, 84, 40, 18, 10, 3328, 3072, 2112, 1280, 720

Offset: 1

Author

Omar E. Pol, Aug 28 2013

Keywords

• nonn
• tabl

Comments

The equivalent sequence for partitions is A138785, see the first comment there.

Examples

			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):
---------------------------------------------------------
. Compositions                   Parts      Sum of parts
.     of 4        Diagram      of size 2     of size 2
---------------------------------------------------------
.                 _ _ _ _
.   1+1+1+1      |_| | | |         0             0
.     2+1+1      |_ _| | |         1             2
.     1+2+1      |_|   | |         1             2
.       3+1      |_ _ _| |         0             0
.     1+1+2      |_| |   |         1             2
.       2+2      |_ _|   |         2             4
.       1+3      |_|     |         0             0
.         4      |_ _ _ _|         0             0
.                                -----        ------
.                           Total  5            10
.
Triangle begins:
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,  12,   7;
320,   288,  192,  112,  60,  30,  14,  8;
704,   640,  432,  256, 140,  72,  35, 16,  9;
1536, 1408,  960,  576, 320, 168,  84, 40, 18, 10;
3328, 3072, 2112, 1280, 720, 384, 196, 96, 45, 20, 11;
...
		

Crossrefs

Column k is k*A045623. Row sums give A001787, n >= 1. Right border gives A000027.

Cf. A001792, A011782, A138785, A221876, A228525, A228527.

Formula

T(n,k) = k*A045623(n-k) = k*A221876(n,k), n >=1, 1<=k<=n.