OFFSET
0,4
COMMENTS
No two adjacent parts of a Carlitz composition are equal.
LINKS
Alois P. Heinz, Rows n = 0..250, flattened
FORMULA
Sum_{k=0..floor(n/3)} (k+1) * T(n,k) = A285994(n) (for n>0).
EXAMPLE
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;
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, expand(
add(`if`(j=i, 0, b(n-j, j)*`if`(j<i, x, 1)), j=1..n)))
end:
T:= n-> (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[j<i, x, 1]], {j, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Apr 27 2014
STATUS
approved