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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). 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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. LINKS Alois P. Heinz, Rows n = 0..150, flattened FORMULA G.f.: 1/(1 - 2*x - 2*(y-1)*x^3/(1 - (y-1)*x - (y-1)*x^2) ). EXAMPLE 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. 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 CROSSREFS Cf. A128588. Sequence in context: A133903 A278263 A236188 * A094752 A214061 A260300 Adjacent sequences:  A242590 A242591 A242592 * A242594 A242595 A242596 KEYWORD nonn,tabf AUTHOR Geoffrey Critzer, May 18 2014 STATUS approved

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Last modified April 8 14:52 EDT 2020. Contains 333314 sequences. (Running on oeis4.)