A372873 Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.

1, 0, 2, 0, 1, 4, 0, 0, 6, 8, 0, 0, 1, 24, 16, 0, 0, 0, 10, 80, 32, 0, 0, 0, 1, 60, 240, 64, 0, 0, 0, 0, 14, 280, 672, 128, 0, 0, 0, 0, 1, 112, 1120, 1792, 256, 0, 0, 0, 0, 0, 18, 672, 4032, 4608, 512, 0, 0, 0, 0, 0, 1, 180, 3360, 13440, 11520, 1024

Offset: 1

Author

Stefano Spezia, May 15 2024

Keywords

• nonn
• tabl

Examples

			The triangle begins:
  1;
  0, 2;
  0, 1, 4;
  0, 0, 6,  8;
  0, 0, 1, 24, 16;
  0, 0, 0, 10, 80,  32;
  0, 0, 0,  1, 60, 240,   64;
  0, 0, 0,  0, 14, 280,  672,  128;
  0, 0, 0,  0,  1, 112, 1120, 1792, 256;
  ...
T(4,3) = 6 since there 6 flattened Catalan words of length 4 with 3 runs of descents: 0010, 0100, 0101, 0110, 0120, and 0121.
		

Links

• Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 11.

Crossrefs

Cf. A000007, A000079, A001788, A007051 (row sums), A033322, A372874.

Programs

Mathematica

T[n_,k_]:=SeriesCoefficient[x*y*(1-2*x*y)/(1-4*x*y-x^2*y+4x^2*y^2),{x,0,n},{y,0,k}]; Table[T[n,k],{n,11},{k,n}]//Flatten (* or *)
T[n_,k_]:=2^(2*k-n-1)*Binomial[n-1, 2*(n-k)]; Table[T[n,k],{n,11},{k,n}]//Flatten

Formula

G.f.: x*y*(1 - 2*x*y)/(1 - 4*x*y - x^2*y + 4*x^2*y^2).

T(n,k) = 2^(2*k-n-1)*binomial(n-1, 2*(n-k)).

T(n,n) = A000079(n-1).

T(n,n-1) = A001788(n-2).

T(n,1) = A000007(n-1).

T(n,2) = A033322(n-1).

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