A372875 Irregular triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k symmetric valleys, with k >= 0.

1, 2, 5, 13, 1, 34, 7, 90, 31, 1, 242, 113, 10, 659, 375, 59, 1, 1808, 1189, 271, 13, 4977, 3686, 1082, 96, 1, 13715, 11284, 3976, 534, 16, 37798, 34239, 13887, 2507, 142, 1, 104154, 103115, 46949, 10555, 929, 19, 286960, 308452, 155200, 41324, 5028, 197, 1

Offset: 1

Author

Stefano Spezia, May 15 2024

Keywords

• nonn
• tabf

Examples

			The irregular triangle begins:
     1;
     2;
     5;
    13,    1;
    34,    7;
    90,   31,    1;
   242,  113,   10;
   659,  375,   59,  1;
  1808, 1189,  271, 13;
  4977, 3686, 1082, 96, 1;
  ...
T(5,1) = 7 since there are 7 flattened Catalan words of length 5 with one symmetric valley: 00101, 01001, 01010, 01011, 01012, 01101, and 01212.
		

Links

• Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See pp. 16-17.

Crossrefs

Cf. A007051 (row sums), A371963, A371964, A372878.

Programs

Mathematica

T[n_,k_]:=SeriesCoefficient[x(1-2x)(1-2x+2x^2-x^2y)/((1-x)(1-5x+8x^2-5x^3-x^2y+2x^3y)),{x,0,n},{y,0,k}];Join[{1},Flatten[ Table[T[n,k],{n,14},{k,0,Floor[(n-2)/2]}]]]

Formula

G.f.: x*(1 - 2*x)*(1 - 2*x + 2*x^2 - x^2*y)/((1 - x)*(1 - 5*x + 8*x^2 - 5*x^3 - x^2*y + 2x^3*y)).

Sum_{k>=0} T(n,k) = A007051(n-1).