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

Author

Emeric Deutsch, Jul 15 2006

Keywords

• nonn
• tabf

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).

Examples

			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;
		

Links

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

Crossrefs

Cf. A000244, A000217, A036290.

Programs

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 (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 *)

Formula

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