A241701 Number T(n,k) of Carlitz compositions of n with exactly k descents; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.

1, 1, 1, 2, 1, 2, 2, 3, 4, 4, 8, 2, 5, 13, 5, 6, 21, 12, 8, 33, 27, 3, 10, 50, 53, 11, 12, 73, 98, 31, 15, 106, 174, 78, 5, 18, 150, 296, 175, 22, 22, 209, 486, 363, 72, 27, 289, 781, 715, 204, 8, 32, 393, 1222, 1342, 510, 43, 38, 529, 1874, 2421, 1168, 159

Offset: 0

Author

Alois P. Heinz, Apr 27 2014

Keywords

• nonn
• tabf

Comments

No two adjacent parts of a Carlitz composition are equal.

Examples

			T(6,0) = 4: [6], [1,5], [2,4], [1,2,3].
T(6,1) = 8: [4,2], [5,1], [3,1,2], [1,3,2], [1,4,1], [2,3,1], [2,1,3], [1,2,1,2].
T(6,2) = 2: [3,2,1], [2,1,2,1].
T(7,0) = 5: [7], [3,4], [1,6], [2,5], [1,2,4].
T(7,1) = 13: [4,3], [6,1], [5,2], [2,1,4], [4,1,2], [1,4,2], [2,3,2], [3,1,3], [1,5,1], [2,4,1], [1,2,3,1], [1,3,1,2], [1,2,1,3].
T(7,2) = 5: [4,2,1], [2,1,3,1], [3,1,2,1], [1,3,2,1], [1,2,1,2,1].
Triangle T(n,k) begins:
00:   1;
01:   1;
02:   1;
03:   2,   1;
04:   2,   2;
05:   3,   4;
06:   4,   8,   2;
07:   5,  13,   5;
08:   6,  21,  12;
09:   8,  33,  27,   3;
10:  10,  50,  53,  11;
11:  12,  73,  98,  31;
12:  15, 106, 174,  78,   5;
13:  18, 150, 296, 175,  22;
14:  22, 209, 486, 363,  72;
15:  27, 289, 781, 715, 204, 8;
		

Links

• Alois P. Heinz, Rows n = 0..250, flattened

Crossrefs

Columns k=0-10 give: A000009, A241691, A241692, A241693, A241694, A241695, A241696, A241697, A241698, A241699, A241700.

Row sums give A003242.

T(3n,n) = A000045(n+1).

T(3n+1,n) = A129715(n) for n>0.

Cf. A238344, A285994.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, expand(
      add(`if`(j=i, 0, b(n-j, j)*`if`(j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..20);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[If[j == i, 0, b[n-j, j]*If[jJean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

Formula

Sum_{k=0..floor(n/3)} (k+1) * T(n,k) = A285994(n) (for n>0).