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A372879
Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k short peak, with k >= 0.
2
1, 2, 4, 1, 9, 5, 22, 18, 1, 56, 58, 8, 145, 178, 41, 1, 378, 532, 173, 11, 988, 1563, 656, 73, 1, 2585, 4535, 2327, 381, 14, 6766, 13030, 7888, 1726, 114, 1, 17712, 37140, 25872, 7124, 709, 17, 46369, 105156, 82758, 27534, 3739, 164, 1, 121394, 296040, 259542, 101350, 17632, 1184, 20
OFFSET
1,2
LINKS
Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 19-20.
FORMULA
G.f.: x*(1 - 2*x)/((1 - x)*(1 - 3*x + x^2*(1 - y))).
T(n,0) = A055588(n-1).
Sum_{k>=0} T(n,k) = A007051(n-1).
EXAMPLE
The irregular triangle begins:
1;
2;
4, 1;
9, 5;
22, 18, 1;
56, 58, 8;
145, 178, 41, 1;
378, 532, 173, 11;
988, 1563, 656, 73, 1;
...
T(6,2) = 8 since there are 8 flattened Catalan words of length 6 with 2 short peaks: 001010, 010100, 010101, 010010, 010120, 010121, 012010, and 012121.
MATHEMATICA
T[n_, k_]:=SeriesCoefficient[x(1-2x)/((1-x)(1-3x+x^2(1-y))), {x, 0, n}, {y, 0, k}]; Table[T[n, k], {n, 14}, {k, 0, Floor[(n-1)/2]}]//Flatten
CROSSREFS
Cf. A007051 (row sums), A055588, A371965, A372883, A372884.
Sequence in context: A344363 A163240 A091958 * A116424 A372883 A135306
KEYWORD
nonn,tabf
AUTHOR
Stefano Spezia, May 15 2024
STATUS
approved