A232435 Number T(n,k) of compositions of n with exactly k (possibly overlapping) occurrences of the consecutive pattern 111; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

1, 1, 2, 3, 1, 7, 0, 1, 13, 2, 0, 1, 24, 5, 2, 0, 1, 46, 11, 4, 2, 0, 1, 89, 21, 11, 4, 2, 0, 1, 170, 45, 23, 11, 4, 2, 0, 1, 324, 99, 47, 23, 12, 4, 2, 0, 1, 618, 209, 102, 52, 23, 13, 4, 2, 0, 1, 1183, 427, 226, 112, 55, 24, 14, 4, 2, 0, 1, 2260, 883, 479

Offset: 0

Author

Alois P. Heinz, Nov 23 2013

Keywords

• nonn
• tabf

Examples

			T(4,0) = 7: [4], [3,1], [2,2], [1,3], [2,1,1], [1,2,1], [1,1,2].
T(7,1) = 11: [4,1,1,1], [2,2,2,1], [1,2,2,2], [1,1,1,4], [1,3,1,1,1], [2,2,1,1,1], [1,1,1,3,1], [2,1,1,1,2], [1,1,1,2,2], [1,1,1,2,1,1], [1,1,2,1,1,1].
T(7,2) = 4: [3,1,1,1,1], [1,1,1,1,3], [1,2,1,1,1,1], [1,1,1,1,2,1].
T(7,3) = 2: [2,1,1,1,1,1], [1,1,1,1,1,2].
T(7,5) = 1: [1,1,1,1,1,1,1].
Triangle T(n,k) begins:
:  0 :   1;
:  1 :   1;
:  2 :   2;
:  3 :   3,  1;
:  4 :   7,  0,  1;
:  5 :  13,  2,  0,  1;
:  6 :  24,  5,  2,  0,  1;
:  7 :  46, 11,  4,  2,  0, 1;
:  8 :  89, 21, 11,  4,  2, 0, 1;
:  9 : 170, 45, 23, 11,  4, 2, 0, 1;
: 10 : 324, 99, 47, 23, 12, 4, 2, 0, 1;
		

Links

• Alois P. Heinz, Rows n = 0..150, flattened

Crossrefs

Column k=0 gives: A128695.

Row sums give: A011782.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1,
      expand(add(`if`(abs(t)<>j, b(n-j, j),
      `if`(t<0, x, 1)*b(n-j, -j)), j=1..n)))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..15);

Mathematica

b[n_, t_] := b[n, t] = If[n==0, 1, Expand[Sum[If[Abs[t] != j, b[n-j, j], If[t<0, x, 1]*b[n-j, -j]], {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, 15}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)