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

0,3

Links

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

Example

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;

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 *)

Crossrefs

Column k=0 gives: A128695.

Row sums give: A011782.

Sequence in context: A130405 A058372 A128264 * A114858 A193491 A364271

Adjacent sequences: A232432 A232433 A232434 * A232436 A232437 A232438

Keyword

nonn,tabf

Author

Alois P. Heinz, Nov 23 2013

Status

approved