A287709 Number of Dyck paths of semilength n such that every peak at level y > 1 is preceded by (at least) one peak at level y-1.

1, 1, 1, 2, 4, 9, 22, 57, 154, 430, 1234, 3625, 10865, 33136, 102598, 321913, 1021963, 3278543, 10617413, 34678693, 114151769, 378436049, 1262822229, 4239469076, 14312153289, 48567846377, 165610404277, 567259571451, 1951218773118, 6738242931451, 23356148951482

Offset

0,4

Comments

Also number of Dyck paths of semilength (n-1) whose maximum height is attained by the initial ascent. (That is, Dyck paths with prefix U^kD, k>=1, and maximum height k.) For a(3)=2: UDUD, UUDD. For a(4)=3: UDUDUD, UUDUDD, UUDDUD, UUUDDD. (Andrei Asinowski and Vít Jelínek) - Andrei Asinowski, Jun 21 2021

References

Andrei Asinowski and Vít Jelínek. Two types of Dyck paths (unpublished manuscript).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Alois P. Heinz, Animation of a(7) = 57 paths

Axel Bacher, Progressive and rushed Dyck paths, arXiv:2403.08120 [math.CO], 2024.

Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 26.

Anthony Guttmann, Analysis of series expansions for non-algebraic singularities, arXiv:1405.5327 [math-ph], 2014.

Wikipedia, Counting lattice paths

Formula

G.f.: 1 + Sum_{k>=0} x^(k+1)/U_{k+1}(1/(2*x)), where U_{k}(x) is the k-th Chebyshev polynomial of the second kind. - Andrei Asinowski, Jun 21 2021

Conjecture: a(n) = Sum_{j=0..n-2} R(n-2, j) for n > 1 with a(0) = a(1) = 1 where R(n, j) = Sum_{p=0..n - j - 1} binomial(j + p, p)*R(n - j - 1, p) for 0 <= j < n with R(n, n) = 1. See A059715 for a similar conjecture. - Mikhail Kurkov, Oct 16 2023

a(n) ~ ((4*Pi)^(5/6) * log(2)^(1/3) / sqrt(3)) * 4^n * exp(-3*(Pi*log(2)/2)^(2/3) * n^(1/3)) * n^(-5/6) [Bacher, 2024, see also Guttmann, 2014, p. 21]. - Vaclav Kotesovec, Mar 14 2024

Example

. a(3) = 2:                 /\
.             /\/\/\     /\/  \     ,
.
. a(4) = 4:                   /\       /\         /\/\
.             /\/\/\/\   /\/\/  \   /\/  \/\   /\/    \   .

Maple

b:= proc(x, y, k) option remember; `if`(x=0, 1,
      `if`(y>0, b(x-1, y-1, max(y, k)), 0)+
      `if`(y<=k and y<x-1, b(x-1, y+1, k), 0))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..35);

Mathematica

b[x_, y_, k_] := b[x, y, k] = If[x == 0, 1, If[y > 0, b[x - 1, y - 1, Max[y, k]], 0] + If[y <= k && y < x - 1, b[x - 1, y + 1, k], 0]];
a[n_] := b[2n, 0, 0];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jun 01 2018, from Maple *)
nmax = 30; CoefficientList[Series[1 + Sum[(Sqrt[x])^(k + 1)/ChebyshevU[k + 1, 1/(2*Sqrt[x])], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, after Andrei Asinowski, Jun 22 2021 *)

Crossrefs

Cf. A000108, A059715, A287776.

Sequence in context: A322504 A099754 A105633 * A348202 A196161 A333069

Adjacent sequences: A287706 A287707 A287708 * A287710 A287711 A287712

Keyword

nonn,changed

Author

Alois P. Heinz, May 30 2017

Status

approved