A119914 Triangle read by rows: T(n,k) is number of ternary words of length n and having k runs of 0's of odd length (0 <= k <= ceiling(n/2); a run of 0's is a subsequence of consecutive 0's of maximal length).

1, 2, 1, 5, 4, 12, 13, 2, 29, 40, 12, 70, 117, 52, 4, 169, 332, 196, 32, 408, 921, 678, 172, 8, 985, 2512, 2216, 768, 80, 2378, 6761, 6952, 3064, 512, 16, 5741, 18004, 21144, 11328, 2640, 192, 13860, 47525, 62762, 39624, 11920, 1424, 32, 33461, 124536

Offset: 0

Author

Emeric Deutsch, May 29 2006

Keywords

• nonn
• tabf

Comments

Row n has 1+ceiling(n/2) terms.

Sum of entries in row n is 3^n (A000244).

T(n,0) = A000129(n+1) (Pell numbers).

T(n,1) = A119915(n).

Sum_{k>=0} k*T(n,k) = A119916(n).

Examples

			T(4,2)=12 because we have 0101, 0102, 0110, 0120, 0201, 0202, 0210, 0220, 1010, 1020, 2010 and 2020.
Triangle starts:
   1;
   2,   1;
   5,   4;
  12,  13,  2;
  29,  40, 12;
  70, 117, 52, 4;
		

Crossrefs

Cf. A000244, A000129, A119915, A119916.

Programs

Maple

G:=(1+t*z)/(1-2*z-z^2-2*t*z^2): Gser:=simplify(series(G,z=0,14)): 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..ceil(n/2)) od; # yields sequence in triangular form

Formula

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

T(n,k) = 2T(n-1,k) + T(n-2,k) + 2T(n-2,k-1) (n >= 2).