A242593 Triangular array read by rows: T(n,k) is the number of length n words on {B,G} that contain exactly k occurrences of the contiguous substrings BGB or GBG. The substrings are allowed to overlap; n>=0, 0<=k<=max(n-2,0).

1, 2, 4, 6, 2, 10, 4, 2, 16, 10, 4, 2, 26, 20, 12, 4, 2, 42, 40, 26, 14, 4, 2, 68, 76, 58, 32, 16, 4, 2, 110, 142, 120, 78, 38, 18, 4, 2, 178, 260, 244, 172, 100, 44, 20, 4, 2, 288, 470, 482, 374, 232, 124, 50, 22, 4, 2, 466, 840, 936, 784, 534, 300, 150, 56, 24, 4, 2, 754, 1488, 1788, 1612, 1176, 726, 376, 178, 62, 26, 4, 2

Offset: 0

Author

Geoffrey Critzer, May 18 2014

Keywords

• nonn
• tabf

Comments

Equivalently, T(n,k) is the number of ways to arrange n children in a line so that exactly k children are in between two children of opposite gender than their own. Children on the ends of the line cannot be counted as "in between".

Row sums = 2^n.

Column k=0 is A128588.

Examples

			Triangle T(n,k) begins:
    1;
    2;
    4;
    6,   2;
   10,   4,   2;
   16,  10,   4,   2;
   26,  20,  12,   4,   2;
   42,  40,  26,  14,   4,  2;
   68,  76,  58,  32,  16,  4,  2;
  110, 142, 120,  78,  38, 18,  4, 2,
  178, 260, 244, 172, 100, 44, 20, 4, 2;
T(4,1) = 4 because we have: BBGB, BGBB, GBGG, GGBG.
T(4,2) = 2 because we have: BGBG, GBGB.
		

Links

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

Crossrefs

Cf. A128588.

Programs

Maple

b:= proc(n, t) option remember; `if`(n=0, 1, expand(
      b(n-1, [4, 3, 4, 4, 3][t])*`if`(t=5, x, 1)+
      b(n-1, [2, 2, 5, 5, 2][t])*`if`(t=3, x, 1)))
    end:
T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n, 1)):
seq(T(n), n=0..16);  # Alois P. Heinz, May 18 2014

Mathematica

nn=10;sol=Solve[{A==va(z^3+z^2A+z B),B==va(z^3+z^2 B + z A)},{A,B}]; Fz[z_,y_]:=Simplify[1/(1-2z-A-B)/.sol/.va->y-1]; Map[Select[#,#>0&]&, Level[CoefficientList[Series[Fz[z,y],{z,0,nn}],{z,y}],{2}]]//Grid

Formula

G.f.: 1/(1 - 2*x - 2*(y-1)*x^3/(1 - (y-1)*x - (y-1)*x^2) ).