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A238279 Triangle read by rows: T(n,k) is the number of compositions of n into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=A004523(n-1). 11
1, 1, 2, 2, 2, 3, 4, 1, 2, 10, 4, 4, 12, 14, 2, 2, 22, 29, 10, 1, 4, 26, 56, 36, 6, 3, 34, 100, 86, 31, 2, 4, 44, 148, 200, 99, 16, 1, 2, 54, 230, 374, 278, 78, 8, 6, 58, 322, 680, 654, 274, 52, 2, 2, 74, 446, 1122, 1390, 814, 225, 22, 1, 4, 88, 573, 1796, 2714, 2058, 813, 136, 10, 4, 88, 778, 2694, 4927 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Same as A238130, with zeros omitted.

Last elements in rows are 1, 1, 2, 2, 1, 4, 2, 1, 6, 2, 1, 8, ... with g.f. -(x^6+x^4-2*x^2-x-1)/(x^6-2*x^3+1).

LINKS

Joerg Arndt and Alois P. Heinz, Rows n = 0..180, flattened

EXAMPLE

Triangle starts:

00:  1;

01:  1;

02:  2;

03:  2,   2;

04:  3,   4,   1;

05:  2,  10,   4;

06:  4,  12,  14,    2;

07:  2,  22,  29,   10,    1;

08:  4,  26,  56,   36,    6;

09:  3,  34, 100,   86,   31,    2;

10:  4,  44, 148,  200,   99,   16,    1;

11:  2,  54, 230,  374,  278,   78,    8;

12:  6,  58, 322,  680,  654,  274,   52,    2;

13:  2,  74, 446, 1122, 1390,  814,  225,   22,   1;

14:  4,  88, 573, 1796, 2714, 2058,  813,  136,  10;

15:  4,  88, 778, 2694, 4927, 4752, 2444,  618,  77,  2;

16:  5, 110, 953, 3954, 8531, 9930, 6563, 2278, 415, 28, 1;

...

Row n=5 is 2, 10, 4 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-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):

seq(T(n), n=0..14);

MATHEMATICA

b[n_, v_] := b[n, v] = If[n == 0, 1, Expand[Sum[b[n-i, i]*If[v == 0 || v == i, 1, x], {i, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, Feb 11 2015, after Maple *)

CROSSREFS

Columns k=0-10 give: A000005 (for n>0), 2*A002133, A244714, A244715, A244716, A244717, A244718, A244719, A244720, A244721, A244722.

Row sums are A011782.

Sequence in context: A124492 A057646 A238892 * A282933 A052275 A244798

Adjacent sequences:  A238276 A238277 A238278 * A238280 A238281 A238282

KEYWORD

nonn,tabf

AUTHOR

Joerg Arndt and Alois P. Heinz, Feb 22 2014

STATUS

approved

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Last modified March 25 15:02 EDT 2019. Contains 321470 sequences. (Running on oeis4.)