A078802 Triangular array T given by T(n,k) = number of 01-words of length n containing k 1's, no three of which are consecutive.

1, 1, 1, 1, 2, 1, 1, 3, 3, 0, 1, 4, 6, 2, 0, 1, 5, 10, 7, 1, 0, 1, 6, 15, 16, 6, 0, 0, 1, 7, 21, 30, 19, 3, 0, 0, 1, 8, 28, 50, 45, 16, 1, 0, 0, 1, 9, 36, 77, 90, 51, 10, 0, 0, 0, 1, 10, 45, 112, 161, 126, 45, 4, 0, 0, 0, 1, 11, 55, 156, 266, 266, 141, 30, 1, 0, 0, 0, 1, 12, 66, 210, 414

Offset: 0

Author

Clark Kimberling, Dec 06 2002

Keywords

• nonn
• tabl

Comments

The rows of T are essentially the antidiagonals of A027907 (trinomial coefficients). Reversing the rows produces A078803. Row sums: A000073.

Also, the diagonals of T are essentially the rows of A027907, so diagonal sums = 3^n. Antidiagonal sums are essentially A060961 (number of ordered partitions of n into 1's, 3's and 5's). - Gerald McGarvey, May 13 2005

Examples

			T(4,3) = 2 counts 1+0+1+1 and 1+1+0+1. Top of triangle T:
  1;
  1, 1;
  1, 2, 1;
  1, 3, 3, 0;
  1, 4, 6, 2, 0;
		

References

• Clark Kimberling, Binary words with restricted repetitions and associated compositions of integers, in Applications of Fibonacci Numbers, vol. 10, Proceedings of the Eleventh International Conference on Fibonacci Numbers and Their Applications, William Webb, editor, Congressus Numerantium, Winnipeg, Manitoba 194 (2009) 141-151.

Crossrefs

Cf. A027907, A078803. See A082601 for another version.

Programs

Maple

seq(seq(sum(binomial(n+1-k,k-j)*binomial(k-j,j),j=0..ceil((k-1)/2)),k=0..n),n=0..20); # Dennis P. Walsh, Apr 04 2012

Mathematica

nn=15; a=1+y x+y^2 x^2;f[list_]:=Select[list,#>0&];Map[f,CoefficientList[Series[a/(1-x a),{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Sep 15 2012 *)

Formula

T(n, k) = T(n-1, k) + T(n-2, k-1) + T(n-3, k-2) with initial values as in first 3 rows.

T(n,k) = Sum_{j=0..ceiling((k-1)/2)} C(n+1-k, k-j)*C(k-j, j). - Dennis P. Walsh, Apr 04 2012

G.f.: (1 + y*x + y^2*x^2)/(1 - (x*(1 + y*x + y^2*x^2))). - Geoffrey Critzer, Sep 15 2012