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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

Row sums are A011782.

LINKS

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

EXAMPLE

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.

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-Fran├žois Alcover, Jan 12 2015, translated from Maple *)

CROSSREFS

Cf. A238279 (same sequence with zeros omitted).

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

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

Sequence in context: A194947 A132339 A137676 * A238707 A181111 A216800

Adjacent sequences:  A238127 A238128 A238129 * A238131 A238132 A238133

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt and Alois P. Heinz, Feb 21 2014

STATUS

approved

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Last modified April 24 00:59 EDT 2017. Contains 285338 sequences.