login
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.
3
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
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt, Apr 27 2013
STATUS
approved