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

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

1,2

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.

Links

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

Example

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.

Crossrefs

Sequence in context: A320382 A259200 A153155 * A134310 A308663 A305713

Adjacent sequences: A225082 A225083 A225084 * A225086 A225087 A225088

Keyword

nonn,tabl

Author

Joerg Arndt, Apr 27 2013

Status

approved