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.

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

Table of n, a(n) for n=1..56.

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

Adjacent sequences: A372876 A372877 A372878 * A372880 A372881 A372882

Keyword

nonn,tabf

Author

Stefano Spezia, May 15 2024

Status

approved