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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.
2
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 Two types of Dyck paths (unpublished manuscript).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
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.
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, May 30 2017
STATUS
approved