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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).

%I A242593 #19 Apr 28 2018 17:23:46
%S A242593 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,
%T A242593 16,4,2,110,142,120,78,38,18,4,2,178,260,244,172,100,44,20,4,2,288,
%U A242593 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
%N 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).
%C A242593 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".
%C A242593 Row sums = 2^n.
%C A242593 Column k=0 is A128588.
%H A242593 Alois P. Heinz, <a href="/A242593/b242593.txt">Rows n = 0..150, flattened</a>
%F A242593 G.f.: 1/(1 - 2*x - 2*(y-1)*x^3/(1 - (y-1)*x - (y-1)*x^2) ).
%e A242593 Triangle T(n,k) begins:
%e A242593     1;
%e A242593     2;
%e A242593     4;
%e A242593     6,   2;
%e A242593    10,   4,   2;
%e A242593    16,  10,   4,   2;
%e A242593    26,  20,  12,   4,   2;
%e A242593    42,  40,  26,  14,   4,  2;
%e A242593    68,  76,  58,  32,  16,  4,  2;
%e A242593   110, 142, 120,  78,  38, 18,  4, 2,
%e A242593   178, 260, 244, 172, 100, 44, 20, 4, 2;
%e A242593 T(4,1) = 4 because we have: BBGB, BGBB, GBGG, GGBG.
%e A242593 T(4,2) = 2 because we have: BGBG, GBGB.
%p A242593 b:= proc(n, t) option remember; `if`(n=0, 1, expand(
%p A242593       b(n-1, [4, 3, 4, 4, 3][t])*`if`(t=5, x, 1)+
%p A242593       b(n-1, [2, 2, 5, 5, 2][t])*`if`(t=3, x, 1)))
%p A242593     end:
%p A242593 T:= n-> (p-> seq(coeff(p,x,i), i=0..degree(p)))(b(n, 1)):
%p A242593 seq(T(n), n=0..16);  # _Alois P. Heinz_, May 18 2014
%t A242593 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
%Y A242593 Cf. A128588.
%K A242593 nonn,tabf
%O A242593 0,2
%A A242593 _Geoffrey Critzer_, May 18 2014