A372883 Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k symmetric peaks, with k >= 0.
1, 2, 4, 1, 9, 5, 23, 17, 1, 63, 51, 8, 176, 149, 39, 1, 491, 439, 153, 11, 1362, 1308, 540, 70, 1, 3762, 3912, 1812, 342, 14, 10369, 11671, 5935, 1439, 110, 1, 28559, 34637, 19175, 5541, 645, 17, 78653, 102222, 61302, 20214, 3170, 159, 1, 216638, 300190, 194080, 71242, 13903, 1089, 20
Offset: 1
Examples
The irregular triangle begins:
1;
2;
4, 1;
9, 5;
23, 17, 1;
63, 51, 8;
176, 149, 39, 1;
491, 439, 153, 11;
1362, 1308, 540, 70, 1;
3762, 3912, 1812, 342, 14;
...
T(4,1) = 5 since there are 5 flattened Catalan words of length 4 with 1 symmetric peak: 0100, 0101, 0010, 0110, and 0121.
Links
- Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 21-22.
Crossrefs
Programs
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Mathematica
T[n_,k_]:=SeriesCoefficient[x(1-x)(1-2x)/(1-5x+8x^2-5x^3-x^2y+2x^3y),{x,0,n},{y,0,k}]; Table[T[n,k],{n,14},{k,0,Floor[(n-1)/2]}]//Flatten
Formula
G.f.: x*(1 - x)*(1 - 2*x)/(1 - 5*x + 8*x^2 - 5*x^3 - x^2*y + 2*x^3*y).
Sum_{k>=0} T(n,k) = A007051(n-1).