A238130 Triangle read by rows: T(n,k) is the number of compositions into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=n.

1, 1, 0, 2, 0, 0, 2, 2, 0, 0, 3, 4, 1, 0, 0, 2, 10, 4, 0, 0, 0, 4, 12, 14, 2, 0, 0, 0, 2, 22, 29, 10, 1, 0, 0, 0, 4, 26, 56, 36, 6, 0, 0, 0, 0, 3, 34, 100, 86, 31, 2, 0, 0, 0, 0, 4, 44, 148, 200, 99, 16, 1, 0, 0, 0, 0, 2, 54, 230, 374, 278, 78, 8, 0, 0, 0, 0, 0, 6, 58, 322, 680, 654, 274, 52, 2, 0, 0, 0, 0, 0

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 21 2014

Keywords

• nonn
• tabl

Comments

First column (k=0) is A000005, second column (k=1) is 2*A002133.

Row sums are A011782.

Examples

			Triangle starts:
00:  1,
01:  1, 0,
02:  2, 0, 0,
03:  2, 2, 0, 0,
04:  3, 4, 1, 0, 0,
05:  2, 10, 4, 0, 0, 0,
06:  4, 12, 14, 2, 0, 0, 0,
07:  2, 22, 29, 10, 1, 0, 0, 0,
08:  4, 26, 56, 36, 6, 0, 0, 0, 0,
09:  3, 34, 100, 86, 31, 2, 0, 0, 0, 0,
10:  4, 44, 148, 200, 99, 16, 1, 0, 0, 0, 0,
11:  2, 54, 230, 374, 278, 78, 8, 0, 0, 0, 0, 0,
12:  6, 58, 322, 680, 654, 274, 52, 2, 0, 0, 0, 0, 0,
13:  2, 74, 446, 1122, 1390, 814, 225, 22, 1, 0, 0, 0, 0, 0,
...
Row 5 is [2, 10, 4, 0, 0, 0] because in the 16 compositions of 5
##:  [composition]  no. of changes
01:  [ 1 1 1 1 1 ]   0
02:  [ 1 1 1 2 ]   1
03:  [ 1 1 2 1 ]   2
04:  [ 1 1 3 ]   1
05:  [ 1 2 1 1 ]   2
06:  [ 1 2 2 ]   1
07:  [ 1 3 1 ]   2
08:  [ 1 4 ]   1
09:  [ 2 1 1 1 ]   1
10:  [ 2 1 2 ]   2
11:  [ 2 2 1 ]   1
12:  [ 2 3 ]   1
13:  [ 3 1 1 ]   1
14:  [ 3 2 ]   1
15:  [ 4 1 ]   1
16:  [ 5 ]   0
there are 2 with no changes, 10 with one change, and 4 with two changes.
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Cf. A238279 (same sequence with zeros omitted).

Cf. A106356 (compositions with k successive parts same).

Cf. A225084 (compositions with maximal up-step k).

Programs

Maple

b:= proc(n, v) option remember; `if`(n=0, 1, expand(
      add(b(n-i, i)*`if`(v=0 or v=i, 1, x), i=1..n)))
    end:
T:= n-> seq(coeff(b(n, 0), x, i), i=0..n):
seq(T(n), n=0..14);

Mathematica

b[n_, v_] := b[n, v] = If[n == 0, 1, Sum[b[n-i, i]*If[v == 0 || v == i, 1, x], {i, 1, n}]]; T[n_] := Table[Coefficient[b[n, 0], x, i], {i, 0, n}]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 12 2015, translated from Maple *)