A120907 Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2} having sum of the lengths of the drops equal to k (n>=0, k>=0). The drops of a ternary word on {0,1,2} are the subwords 10,20 and 21, their lengths being the differences 1, 2 and 1, respectively.

1, 3, 6, 2, 1, 10, 10, 7, 15, 30, 31, 4, 1, 21, 70, 105, 36, 11, 28, 140, 294, 184, 76, 6, 1, 36, 252, 714, 696, 396, 78, 15, 45, 420, 1554, 2160, 1666, 566, 141, 8, 1, 55, 660, 3102, 5808, 5918, 2990, 995, 136, 19, 66, 990, 5775, 13992, 18348, 12746, 5615, 1280, 226

Offset: 0

Author

Emeric Deutsch, Jul 15 2006

Keywords

• nonn
• tabf

Comments

Row n has 2*floor(n/2)+1 terms (i.e. each of the rows 2n and 2n+1 has 2n+1 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)=4(n-1)3^(n-2)=A120908(n)=4*A027471(n).

Examples

			T(4,3)=4 because we have 1020,2010,2021 and 2120.
Triangle starts:
1;
3;
6,2,1;
10,10,7;
15,30,31,4,1;
21,70,105,36,11;
		

Crossrefs

Cf. A000244, A000217, A120908, A027471, A120906.

Programs

Maple

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

Formula

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