A371963 a(n) is the sum of all valleys in the set of Catalan words of length n.

0, 0, 0, 0, 1, 8, 44, 209, 924, 3927, 16303, 66691, 270181, 1087371, 4356131, 17394026, 69289961, 275543036, 1094352236, 4342295396, 17218070066, 68239187876, 270351828476, 1070824260326, 4240695090452, 16792454677874, 66492351226050, 263285419856250, 1042540731845950

Offset: 0

Author

Stefano Spezia, Apr 14 2024

Keywords

• nonn

Examples

			a(4) = 1 because there is 1 Catalan word of length 4 with one valley: 0101.
a(5) = 8 because there are 8 Catalan words of length 5 with one valley: 00101, 01010, 01011, 01012, 01101, 01201, and 01212 (see Figure 9 at p. 14 in Baril et al.).
		

Links

• Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, Enumerating runs, valleys, and peaks in Catalan words, arXiv:2404.05672 [math.CO], 2024. See Corollary 4.5, p. 15.

Crossrefs

Cf. A371964, A371965.

Cf. A003516.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, 0,
      a(n-1)+binomial(2*n-3, n-4))
    end:
seq(a(n), n=0..28);  # Alois P. Heinz, Apr 15 2024

Mathematica

CoefficientList[Series[(1 - 5x+5x^2-(1-3x+x^2)Sqrt[1-4x])/(2(1-x)x Sqrt[1-4x]),{x,0,28}],x]

Python

from math import comb
def A371963(n): return sum(comb((n-i<<1)-3,n-i-4) for i in range(n-3)) # Chai Wah Wu, Apr 15 2024

Formula

G.f.: (1-5*x+5*x^2-(1-3*x+x^2)*sqrt(1-4*x))/(2*(1-x)*x*sqrt(1-4*x)).

a(n) = Sum_{i=1..n-1} binomial(2*(n-i)-1,n-i-3).

a(n) ~ 2^(2*n)/(6*sqrt(Pi*n)).

a(n) - a(n-1) = A003516(n-2).