A287709 - OEIS

%I #56 Jan 25 2025 12:32:08

%S 1,1,1,2,4,9,22,57,154,430,1234,3625,10865,33136,102598,321913,

%T 1021963,3278543,10617413,34678693,114151769,378436049,1262822229,

%U 4239469076,14312153289,48567846377,165610404277,567259571451,1951218773118,6738242931451,23356148951482

%N 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.

%C 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

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 Two types of Dyck paths (unpublished manuscript).

%H Alois P. Heinz, <a href="/A287709/b287709.txt">Table of n, a(n) for n = 0..1000</a>

%H Alois P. Heinz, <a href="/A287709/a287709.gif">Animation of a(7) = 57 paths</a>

%H Axel Bacher, <a href="https://arxiv.org/abs/2403.08120">Progressive and rushed Dyck paths</a>, arXiv:2403.08120 [math.CO], 2024.

%H Manosij Ghosh Dastidar and Michael Wallner, <a href="https://arxiv.org/abs/2402.17849">Bijections and congruences involving lattice paths and integer compositions</a>, arXiv:2402.17849 [math.CO], 2024. See p. 26.

%H Anthony Guttmann, <a href="https://arxiv.org/abs/1405.5327">Analysis of series expansions for non-algebraic singularities</a>, arXiv:1405.5327 [math-ph], 2014.

%H Toufik Mansour and Mark Shattuck, <a href="https://math.colgate.edu/~integers/z5/z5.pdf">Enumeration of Catalan and smooth words according to capacity</a>, Integers (2025) Vol. 25, Art. No. A5. See p. 12.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

%F 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

%F 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

%F 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

%e . a(3) = 2:                 /\

%e .             /\/\/\     /\/  \     ,

%e .

%e . a(4) = 4:                   /\       /\         /\/\

%e .             /\/\/\/\   /\/\/  \   /\/  \/\   /\/    \   .

%p b:= proc(x, y, k) option remember; `if`(x=0, 1,

%p       `if`(y>0, b(x-1, y-1, max(y, k)), 0)+

%p       `if`(y<=k and y<x-1, b(x-1, y+1, k), 0))

%p     end:

%p a:= n-> b(2*n, 0$2):

%p seq(a(n), n=0..35);

%t 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]];

%t a[n_] := b[2n, 0, 0];

%t Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 01 2018, from Maple *)

%t 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 *)

%Y Cf. A000108, A059715, A287776.

%K nonn,changed

%O 0,4

%A _Alois P. Heinz_, May 30 2017