A232396 Triangular array read by rows: T(n,k) is the number of compositions of n with no two consecutive identical parts that have exactly k parts = 1, n>=0, 0<=k<=ceiling(n/3).

1, 0, 1, 1, 0, 1, 2, 1, 2, 1, 3, 3, 1, 3, 8, 3, 6, 9, 7, 1, 7, 20, 10, 2, 14, 27, 25, 5, 18, 52, 39, 14, 1, 30, 77, 78, 26, 3, 45, 132, 133, 60, 8, 66, 213, 240, 117, 24, 1, 107, 334, 421, 232, 54, 4, 157, 562, 716, 450, 127, 12, 245, 872, 1265, 842, 279, 38, 1

Offset: 0

Author

Geoffrey Critzer, Nov 23 2013

Keywords

• nonn
• tabf

Examples

			1;
0,  1;
1,  0;
1,  2;
1,  2,  1;
3,  3,  1;
3,  8,  3;
6,  9,  7,  1;
7,  20, 10, 2;
14, 27, 25, 5;
18, 52, 39, 14, 1;
T(7,2) = 7 because we have: 1+2+1+3, 1+2+3+1, 1+3+1+2, 1+3+2+1, 1+5+1, 2+1+3+1, 3+1+2+1.
		

Links

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

Crossrefs

Row sums give: A003242.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, expand(
      add(`if`(j=t, 0, b(n-j, j)*`if`(j=1, x, 1)), j=1..n)))
    end:
T:= n-> seq(coeff(b(n, 0), x, i), i=0..ceil(n/3)):
seq(T(n), n=0..20); # Alois P. Heinz, Nov 24 2013

Mathematica

nn=10;CoefficientList[Series[1/(1- u z/(1+ u z) - Sum[z^j/(1+z^j),{j,2,nn}]),{z,0,nn}],{z,u}]//Grid

Formula

G.f.: 1/( 1 - y*x/(1 + y*x) - Sum_{j>=2} x^j/(1 + x^j) ).