A371964 a(n) is the sum of all symmetric valleys in the set of Catalan words of length n.

0, 0, 0, 0, 1, 7, 35, 155, 650, 2652, 10660, 42484, 168454, 665874, 2627130, 10353290, 40775045, 160534895, 631970495, 2487938015, 9795810125, 38576953505, 151957215305, 598732526105, 2359771876175, 9303298456451, 36688955738099, 144732209103699, 571117191135799

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 symmetric valley: 0101.
a(5) = 7 because there are 7 Catalan words of length 5 with one symmetric valley: 00101, 01001, 01010, 01011, 01012, 01101, and 01212 (see example at p. 16 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 4.7, pp. 16-17.

Crossrefs

Cf. A371963, A371965.

Cf. A000108, A002694, A079309.

Programs

Maple

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

Mathematica

CoefficientList[Series[(1-4x+2x^2-(1-2x)Sqrt[1-4x])/(2(1-x) Sqrt[1-4x]),{x,0,29}],x]

Python

from math import comb
def A371964(n): return sum(comb((n-i<<1)-4,n-i-4) for i in range(n-3)) # Chai Wah Wu, Apr 15 2024

Formula

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

a(n) = (3*n - 2)*A000108(n-1) - A079309(n) for n > 0.

a(n) ~ 2^(2*n)/(12*sqrt(Pi*n)).

a(n)/A371963(n) ~ 1/2.

a(n) - a(n-1) = A002694(n-2).