A118357 Triangle read by rows: T(n,k) is the number of ternary sequences of length n containing k subsequences 00 (n>=0, 0<=k<=max(0,n-1)).

1, 3, 8, 1, 22, 4, 1, 60, 16, 4, 1, 164, 56, 18, 4, 1, 448, 188, 68, 20, 4, 1, 1224, 608, 248, 80, 22, 4, 1, 3344, 1920, 864, 312, 92, 24, 4, 1, 9136, 5952, 2928, 1152, 380, 104, 26, 4, 1, 24960, 18192, 9696, 4128, 1472, 452, 116, 28, 4, 1, 68192, 54976, 31536, 14400

Offset

0,2

Comments

Sum of entries in row n is 3^n (A000244). T(n,0) = A028859(n). T(n,1) = A073388(n-2). Sum(k*T(n,k),k=0..n-1) = (n-1)*3^(n-2) (A027471).

Links

Alois P. Heinz, n = 0..141, flattened

Formula

G.f.: G-1, where G = G(t,z) = [1+(1-t)z]/[1-(2+t)z-2(1-t)z^2]. G.f. of column k is z^(k+1)*(1-2z)^(k-1)/(1-2z-2z^2)^(k+1) (k>=1).

Example

T(4,2) = 4 because we have 0001, 0002, 1000 and 2000.
Triangle starts:
1;
3;
8,1;
22,4,1;
60,16,4,1;

Maple

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

Mathematica

nn=15; a=1/(1-2x); b=x/(1-y x)+1; f[list_]:=Select[list, #>0&]; Map[f, CoefficientList[Series[a b/(1-2x^2/((1-y x)(1-2x))), {x, 0, nn}], {x, y}]]//Grid  (* Geoffrey Critzer, Nov 19 2012 *)

Crossrefs

Cf. A000244, A028859, A073388, A027471.

Sequence in context: A288875 A152230 A181371 * A278866 A281287 A278859

Adjacent sequences: A118354 A118355 A118356 * A118358 A118359 A118360

Keyword

nonn,tabf

Author

Emeric Deutsch, May 24 2006

Status

approved