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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.

%I A287709 #60 Feb 11 2025 10:31:39
%S A287709 1,1,1,2,4,9,22,57,154,430,1234,3625,10865,33136,102598,321913,
%T A287709 1021963,3278543,10617413,34678693,114151769,378436049,1262822229,
%U A287709 4239469076,14312153289,48567846377,165610404277,567259571451,1951218773118,6738242931451,23356148951482
%N 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.
%C A287709 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 A287709 Andrei Asinowski and Vít Jelínek. Two types of Dyck paths (unpublished manuscript).
%H A287709 Alois P. Heinz, <a href="/A287709/b287709.txt">Table of n, a(n) for n = 0..1000</a>
%H A287709 Andrei Asinowski and Michaela A. Polley, <a href="https://arxiv.org/abs/2501.11781">Patterns in rectangulations. Part I: T-like patterns, inversion sequence classes I(010, 101, 120, 201) and I(011, 201), and rushed Dyck paths</a>, arXiv:2501.11781 [math.CO], 2025. See pp. 1, 7, 25, 27.
%H A287709 Axel Bacher, <a href="https://arxiv.org/abs/2403.08120">Progressive and rushed Dyck paths</a>, arXiv:2403.08120 [math.CO], 2024.
%H A287709 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 A287709 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 A287709 Alois P. Heinz, <a href="/A287709/a287709.gif">Animation of a(7) = 57 paths</a>
%H A287709 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 A287709 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A287709 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 A287709 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 A287709 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 A287709 . a(3) = 2:                 /\
%e A287709 .             /\/\/\     /\/  \     ,
%e A287709 .
%e A287709 . a(4) = 4:                   /\       /\         /\/\
%e A287709 .             /\/\/\/\   /\/\/  \   /\/  \/\   /\/    \   .
%p A287709 b:= proc(x, y, k) option remember; `if`(x=0, 1,
%p A287709       `if`(y>0, b(x-1, y-1, max(y, k)), 0)+
%p A287709       `if`(y<=k and y<x-1, b(x-1, y+1, k), 0))
%p A287709     end:
%p A287709 a:= n-> b(2*n, 0$2):
%p A287709 seq(a(n), n=0..35);
%t A287709 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 A287709 a[n_] := b[2n, 0, 0];
%t A287709 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Jun 01 2018, from Maple *)
%t A287709 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 A287709 Cf. A000108, A059715, A287776.
%K A287709 nonn
%O A287709 0,4
%A A287709 _Alois P. Heinz_, May 30 2017