A225085 Triangle read by rows: T(n,k) is the number of compositions of n with maximal up-step <= k; n>=1, 0<=k

1, 2, 2, 3, 4, 4, 5, 7, 8, 8, 7, 13, 15, 16, 16, 11, 23, 29, 31, 32, 32, 15, 41, 55, 61, 63, 64, 64, 22, 72, 105, 119, 125, 127, 128, 128, 30, 127, 199, 233, 247, 253, 255, 256, 256, 42, 222, 378, 455, 489, 503, 509, 511, 512, 512, 56, 388, 716, 889, 967, 1001, 1015, 1021, 1023, 1024, 1024

Offset: 1

Author

Joerg Arndt, Apr 27 2013

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of compositions [p(1), p(2), ..., p(k)] of n such that max(p(j) - p(j-1)) <= k.

Rows are partial sums of rows of A225084.

The first column is A000041 (partition numbers), the second column is A003116, and the third column is A224959.

The diagonal is A011782.

Examples

			Triangle begins
01: 1,
02: 2, 2,
03: 3, 4, 4,
04: 5, 7, 8, 8,
05: 7, 13, 15, 16, 16,
06: 11, 23, 29, 31, 32, 32,
07: 15, 41, 55, 61, 63, 64, 64,
08: 22, 72, 105, 119, 125, 127, 128, 128,
09: 30, 127, 199, 233, 247, 253, 255, 256, 256,
10: 42, 222, 378, 455, 489, 503, 509, 511, 512, 512,
...
The fifth row corresponds to the following statistics:
#:  M   composition
01:  0  [ 1 1 1 1 1 ]
02:  1  [ 1 1 1 2 ]
03:  1  [ 1 1 2 1 ]
04:  2  [ 1 1 3 ]
05:  1  [ 1 2 1 1 ]
06:  1  [ 1 2 2 ]
07:  2  [ 1 3 1 ]
08:  3  [ 1 4 ]
09:  0  [ 2 1 1 1 ]
10:  1  [ 2 1 2 ]
11:  0  [ 2 2 1 ]
12:  1  [ 2 3 ]
13:  0  [ 3 1 1 ]
14:  0  [ 3 2 ]
15:  0  [ 4 1 ]
16:  0  [ 5 ]
There are 7 compositions with no up-step (M<=0), 13 with M<=1, 15 with M<=2, 16 with M<=3, and 16 with M<=4.
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 1..141, flattened