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

0,6

Links

Table of n, a(n) for n=0..28.

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.

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).

Example

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.).

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]

Prog

(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

Crossrefs

Cf. A371963, A371965.

Cf. A000108, A002694, A079309.

Sequence in context: A217274 A000588 A005285 * A006095 A171477 A265612

Adjacent sequences: A371961 A371962 A371963 * A371965 A371966 A371967

Keyword

nonn

Author

Stefano Spezia, Apr 14 2024

Status

approved