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 A118424 Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 001 (n,k>=0). 3
 1, 2, 4, 7, 1, 12, 4, 20, 12, 33, 30, 1, 54, 68, 6, 88, 144, 24, 143, 291, 77, 1, 232, 568, 216, 8, 376, 1080, 552, 40, 609, 2012, 1318, 156, 1, 986, 3688, 2988, 520, 10, 1596, 6672, 6504, 1552, 60, 2583, 11941, 13702, 4266, 275, 1, 4180, 21180, 28104, 11000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Row n has 1+floor(n/3) terms. Sum of entries in row n is 2^n (A000079). T(n,0)=A000071(n+3)=fibonacci(n+3)-1. T(n,1)=A118425(n). Sum(k*T(n,k),k=0..n-1)=(n-2)*2^(n-3) (A001787). LINKS Alois P. Heinz, Rows n = 0..250, flattened FORMULA G.f.: G(t,z) = 1/[1-2z+(1-t)z^3]. Recurrence relation: T(n,k) = 2T(n-1,k) -T(n-3,k) +T(n-3,k-1) for n>=3. EXAMPLE T(7,2) = 6 because we have 0bb, 1bb, b0b, b1b, bb0 and bb1, where b=001. Triangle starts: 1; 2; 4; 7, 1; 12, 4; 20, 12; 33, 30, 1; MAPLE G:=1/(1-2*z+(1-t)*z^3): Gser:=simplify(series(G, z=0, 20)): P[0]:=1: for n from 1 to 17 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 17 do seq(coeff(P[n], t, j), j=0..floor(n/3)) od; # yields sequence in triangular form MATHEMATICA nn=15; Map[Select[#, #>0&]&, CoefficientList[Series[1/(1-2z-(u-1)z^3), {z, 0, nn}], {z, u}]]//Grid (* Geoffrey Critzer, Dec 03 2013 *) CROSSREFS Cf. A000071, A000079, A001787, A118425. Sequence in context: A061501 A272001 A089349 * A118429 A110317 A098073 Adjacent sequences: A118421 A118422 A118423 * A118425 A118426 A118427 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Apr 27 2006 STATUS approved

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Last modified July 22 12:22 EDT 2024. Contains 374499 sequences. (Running on oeis4.)