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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.
3
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
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
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jul 15 2006
STATUS
approved