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

1,2

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. 21-22.

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).

Example

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.

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

Crossrefs

Cf. A007051 (row sums), A290900 (2nd column), A369328 (1st column), A371965, A372879, A372884.

Sequence in context: A091958 A372879 A116424 * A135306 A242352 A270953

Adjacent sequences: A372880 A372881 A372882 * A372884 A372885 A372886

Keyword

nonn,tabf

Author

Stefano Spezia, May 15 2024

Status

approved