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

0,2

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).

Links

Table of n, a(n) for n=0..58.

Formula

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

Example

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;

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

Crossrefs

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

Sequence in context: A255051 A145896 A159963 * A133358 A058099 A292789

Adjacent sequences: A120904 A120905 A120906 * A120908 A120909 A120910

Keyword

nonn,tabf

Author

Emeric Deutsch, Jul 15 2006

Status

approved