A225084 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, 0, 3, 1, 0, 5, 2, 1, 0, 7, 6, 2, 1, 0, 11, 12, 6, 2, 1, 0, 15, 26, 14, 6, 2, 1, 0, 22, 50, 33, 14, 6, 2, 1, 0, 30, 97, 72, 34, 14, 6, 2, 1, 0, 42, 180, 156, 77, 34, 14, 6, 2, 1, 0, 56, 332, 328, 173, 78, 34, 14, 6, 2, 1, 0, 77, 600, 681, 378, 177, 78, 34, 14, 6, 2, 1, 0, 101, 1078, 1393, 818, 393, 178, 78, 34, 14, 6, 2, 1, 0

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.

The first column is A000041 (partition numbers).

Sum of first and second column is A003116.

Sum of the first three columns is A224959.

The second columns deviates from A054454 after the term 600.

Row sums are A011782.

Examples

			Triangle starts:
01: 1,
02: 2, 0,
03: 3, 1, 0,
04: 5, 2, 1, 0,
05: 7, 6, 2, 1, 0,
06: 11, 12, 6, 2, 1, 0,
07: 15, 26, 14, 6, 2, 1, 0,
08: 22, 50, 33, 14, 6, 2, 1, 0,
09: 30, 97, 72, 34, 14, 6, 2, 1, 0,
10: 42, 180, 156, 77, 34, 14, 6, 2, 1, 0,
11: 56, 332, 328, 173, 78, 34, 14, 6, 2, 1, 0,
12: 77, 600, 681, 378, 177, 78, 34, 14, 6, 2, 1, 0,
13: 101, 1078, 1393, 818, 393, 178, 78, 34, 14, 6, 2, 1, 0,
14: 135, 1917, 2821, 1746, 863, 397, 178, 78, 34, 14, 6, 2, 1, 0,
15: 176, 3393, 5660, 3695, 1872, 877, 398, 178, 78, 34, 14, 6, 2, 1, 0,
...
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), 6 with M=1, 2 with M=2, and 1 with M=3.
		

Links

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

Crossrefs

Cf. A225085 (partial sums of rows).

T(2n,n) gives A059570.

Programs

Maple

b:= proc(n, v) option remember; `if`(n=0, 1, add((p->
      `if`(i seq(coeff(b(n, 0), x, i), i=0..n-1):
seq(T(n), n=1..14);  # Alois P. Heinz, Feb 22 2014

Mathematica

b[n_, v_] := b[n, v] = If[n == 0, 1, Sum[Function[{p}, If[iJean-François Alcover, Feb 18 2015, after Alois P. Heinz *)