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A242593
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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).
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2
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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G.f.: 1/(1 - 2*x - 2*(y-1)*x^3/(1 - (y-1)*x - (y-1)*x^2) ).
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EXAMPLE
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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.
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MAPLE
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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)):
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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