A120906 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2} having k drops (n>=0, k>=0). The drops of a ternary word on {0,1,2} are the subwords 10,20 and 21.

1, 3, 6, 3, 10, 16, 1, 15, 51, 15, 21, 126, 90, 6, 28, 266, 357, 77, 1, 36, 504, 1107, 504, 36, 45, 882, 2907, 2304, 414, 9, 55, 1452, 6765, 8350, 2850, 210, 1, 66, 2277, 14355, 25653, 14355, 2277, 66, 78, 3432, 28314, 69576, 58278, 16236, 1221, 12, 91, 5005

Offset

0,2

Comments

Row n has 1+floor(2n/3) terms. Row sums are the powers of 3 (A000244). T(n,0)=A000217(n+1) (the triangular numbers). Sum(k*T(n,k),k>=0)=(n-1)*3^(n-1)=A036290(n-1).

Links

Alois P. Heinz, Rows n = 0..200, flattened

Formula

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

Example

T(5,3) = 6 because we have 1/02/1/0, 2/02/1/0, 2/1/01/0, 2/1/02/0, 2/12/1/0 and 2/1/02/1, the middle points of the drops being indicated by /.
Triangle starts:
1;
3;
6,    3;
10,  16,  1;
15,  51, 15;
21, 126, 90, 6;

Maple

G:=1/((1-z)^3-3*t*z^2+2*t*z^3-t^2*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: for n from 0 to 12 do seq(coeff(P[n], t, j), j=0..floor(2*n/3)) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, expand(
      add(b(n-1, j)*`if`(j<i, x, 1), j=0..2)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..15);  # Alois P. Heinz, May 19 2014

Mathematica

sol=Solve[{a==v z^2, b==v z^2, c==v(z^2+a z)}, {a, b, c}]; f[z_, u_]:=1/(1-3z-a-b-c)/.sol/.v->u-1; nn=10; Map[Select[#, #>0&]&, Level[CoefficientList[Series[f[z, u], {z, 0, nn}], {z, u}], {2}]]//Grid (* Geoffrey Critzer, May 19 2014 *)

Crossrefs

Cf. A000244, A000217, A036290.

Sequence in context: A128503 A210193 A368952 * A258758 A210201 A275535

Adjacent sequences: A120903 A120904 A120905 * A120907 A120908 A120909

Keyword

nonn,tabf

Author

Emeric Deutsch, Jul 15 2006

Status

approved