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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-François 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.)