OFFSET
1,2
COMMENTS
With offset 0 for the variable k, T(n,k) is the number of flattened Catalan words of length n with exactly k peaks. In such case, T(4,1) = 6 corresponds to 6 flattened Catalan words of length 4 with 1 peak: 0010, 0100, 0110, 0101, 0120, and 0121. See Baril et al. at page 20.
LINKS
Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 8-9.
FORMULA
EXAMPLE
The irregular triangle begins:
1;
2;
4, 1;
8, 6;
16, 24, 1;
32, 80, 10;
64, 240, 60, 1;
128, 672, 280, 14;
256, 1792, 1120, 112, 1;
...
T(4,2) = 6 since there are 6 flattened Catalan words of length 4 with 2 runs of weak ascents: 0010, 0100, 0101, 0110, 0120, and 0121.
MATHEMATICA
T[n_, k_]:=SeriesCoefficient[(1-2x)*x*y/(1-4*x+4*x^2-x^2*y), {x, 0, n}, {y, 0, k}]; Table[T[n, k], {n, 14}, {k, Ceiling[n/2]}] //Flatten (* or *)
T[n_, k_]:=2^(n-2k+1)Binomial[n-1, 2k-2]; Table[T[n, k], {n, 14}, {k, Ceiling[n/2]}]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Stefano Spezia, May 15 2024
STATUS
approved