%I
%S 1,1,3,8,23,71,229,759,2566,8817,30717,108278,385509,1384262,5006925,
%T 18225400,66711769,245400354,906711758,3363516354,12522302087,
%U 46773419089,175232388955,658295899526,2479268126762,9359152696924
%N Number of Dyck paths of semilength n having exactly one peak of maximum height.
%C Also number of peaks of maximum height in all Dyck paths of semilength n1. Example: a(3)=3 because in (UD)(UD) and U(UD)D we have three peaks of maximum height (shown between parentheses).
%H Miklos Bona, Elijah DeJonge, <a href="https://arxiv.org/abs/2003.10640">Pattern avoiding permutations and involutions with a unique longest increasing subsequence</a>, arXiv:2003.10640 [math.CO], 2020.
%H Miklós Bóna, Elijah DeJonge, <a href="https://permutationpatterns.com/files/2020/06/WednesdayB_Bona.pdf">Pattern Avoiding Permutations and Involutions with a Unique Longest Increasing Subsequence</a>, (2020).
%F G.f.: g(z) = Sum_{j>=1} z^j/f(j)^2, where the f(j)'s are the Fibonacci polynomials (in z) defined by f(0)=f(1)=1, f(j)=f(j1)zf(j2), j>=2.
%F a(n) = A152879(n,1).
%F a(n) = Sum_{k=1..n} k*A152879(n1,k).
%e a(3)=3 because we have UU(UD)DD, UDU(UD)D, U(UD)DUD, where U=(1,1), D=(1,1), with the peak of maximum height shown between parentheses; the path UUDUDD does not qualify because it has two peaks of maximum height.
%p f[0] := 1: f[1] := 1: for i from 2 to 35 do f[i] := sort(expand(f[i1]z*f[i2])) end do; g := sum(z^j/f[j]^2, j = 1 .. 34): gser := series(g, z = 0, 30): seq(coeff(gser, z, n), n = 1 .. 27);
%Y Cf. A152879.
%K nonn
%O 1,3
%A _Emeric Deutsch_, Jan 02 2009
