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A118890
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Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0110 (n,k >= 0).
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4
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1, 2, 4, 8, 15, 1, 28, 4, 52, 12, 97, 30, 1, 181, 70, 5, 338, 156, 18, 631, 339, 53, 1, 1178, 722, 142, 6, 2199, 1515, 357, 25, 4105, 3140, 862, 84, 1, 7663, 6444, 2018, 252, 7, 14305, 13116, 4614, 700, 33, 26704, 26513, 10348, 1846, 124, 1, 49850, 53280, 22844
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OFFSET
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0,2
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COMMENTS
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Row n has ceiling(n/3) terms (n>=1).
Sum of entries in row n is 2^n (A000079).
Sum_{n>=0} k*T(n,k) = (n-3)*2^(n-4) (A001787).
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LINKS
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FORMULA
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G.f.: G(t,z) = (1+(1-t)z^3)/(1 - 2z + (1-t)(1-z)z^3).
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EXAMPLE
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T(8,2) = 5 because we have 01100110, 01101100, 01101101, 00110110 and 10110110.
Triangle starts:
1;
2;
4;
8;
15, 1;
28, 4;
52, 12;
97, 30, 1;
181, 70, 5;
338, 156, 18;
631, 339, 53, 1;
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MAPLE
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G:=(1+(1-t)*z^3)/(1-2*z+(1-t)*(1-z)*z^3): Gser:=simplify(series(G, z=0, 24)): P[0]:=1: for n from 1 to 18 do P[n]:=sort(coeff(Gser, z^n)) od: 1; for n from 1 to 18 do seq(coeff(P[n], t, j), j=0..ceil(n/3)-1) od; # yields sequence in triangular form
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MATHEMATICA
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nn=18; c=x^3; Map[Select[#, #>0&]&, CoefficientList[Series[1/(1-2x - (y-1)x^4/ (1-(y-1)c)), {x, 0, nn}], {x, y}]]//Flatten (* Geoffrey Critzer, Dec 25 2013 *)
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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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