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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.
3
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
0,5
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
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.
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
EXAMPLE
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;
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 *)
CROSSREFS
Cf. A027907, A078803. See A082601 for another version.
Sequence in context: A053423 A216201 A127514 * A216232 A217765 A237928
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Dec 06 2002
STATUS
approved