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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). 32
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).

For n > 0, also the number of compositions of n with k + 1 runs. - Gus Wiseman, Apr 10 2020

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 *)

Table[If[n==0, 1, Length[Select[Join@@Permutations/@IntegerPartitions[n], Length[Split[#]]==k+1&]]], {n, 0, 12}, {k, 0, If[n==0, 0, Floor[2*(n-1)/3]]}] (* Gus Wiseman, Apr 10 2020 *)

CROSSREFS

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

Row lengths are A004523.

Row sums are A011782.

The version counting adjacent equal parts is A106356.

The version for ascents/descents is A238343.

The version for weak ascents/descents is A333213.

The k-th composition in standard-order has A124762(k) adjacent equal parts, A124767(k) maximal runs, A333382(k) adjacent unequal parts, and A333381(k) maximal anti-runs.

Cf. A064113, A333214, A333216.

Sequence in context: A124492 A057646 A238892 * A282933 A328576 A052275

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 May 27 18:56 EDT 2020. Contains 334664 sequences. (Running on oeis4.)