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

1,2

Comments

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

Links

Table of n, a(n) for n=1..60.

Formula

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

Example

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.

Sequence in context: A283235 A209763 A209761 * A209745 A249620 A358537

Adjacent sequences: A228523 A228524 A228525 * A228527 A228528 A228529

Keyword

nonn,tabl

Author

Omar E. Pol, Aug 28 2013

Status

approved