A119825 Triangle read by rows: T(n,k) is the number of ternary sequences of length n containing k subsequences 000 (consecutively; n,k>=0).

1, 3, 9, 26, 1, 76, 4, 1, 222, 16, 4, 1, 648, 60, 16, 4, 1, 1892, 212, 62, 16, 4, 1, 5524, 728, 224, 64, 16, 4, 1, 16128, 2444, 788, 236, 66, 16, 4, 1, 47088, 8064, 2712, 848, 248, 68, 16, 4, 1, 137480, 26256, 9168, 2984, 908, 260, 70, 16, 4, 1, 401392, 84576, 30576

Offset: 0

Author

Emeric Deutsch, May 26 2006

Keywords

• nonn
• tabf

Comments

Rows 0 and 1 have one term each; row n (n>=2) have n-1 terms. Sum of entries in row n is 3^n (A000244). T(n,0) = A119826(n) T(n,1) = A119827(n) Sum(k*T(n,k), k>=0)=(n-2)*3^(n-3) = A027741(n-1).

Examples

			T(5,2) = 4 because we have 00001, 00002, 10000 and 20000.
Triangle starts:
    1;
    3;
    9;
   26,  1;
   76,  4, 1;
  222, 16, 4, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..150, flattened

Crossrefs

Cf. A000244, A119826, A119827, A027741.

Programs

Maple

G:=(1+(1-t)*z+(1-t)*z^2)/(1-(2+t)*z-2*(1-t)*z^2-2*(1-t)*z^3): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser,z^n)) od: 1;3;for n from 2 to 12 do seq(coeff(P[n],t,j),j=0..n-2) od; # yields sequence in triangular form

Mathematica

nn=10; f[list_]:=Select[list,#>0&]; a=x^2/(1-y x) +x; Map[f,CoefficientList[Series[(a+1)/(1-2x-2x a),{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Oct 31 2012 *)

Formula

G.f.: G(t,z)=[1+(1-t)z+(1-t)z^2]/[1-(2+t)z-2(1-t)z^2-2(1-t)z^3].