

A152880


Number of Dyck paths of semilength n having exactly one peak of maximum height.


1



1, 1, 3, 8, 23, 71, 229, 759, 2566, 8817, 30717, 108278, 385509, 1384262, 5006925, 18225400, 66711769, 245400354, 906711758, 3363516354, 12522302087, 46773419089, 175232388955, 658295899526, 2479268126762, 9359152696924
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OFFSET

1,3


COMMENTS

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).


LINKS

Table of n, a(n) for n=1..26.
Miklos Bona, Elijah DeJonge, Pattern avoiding permutations and involutions with a unique longest increasing subsequence, arXiv:2003.10640 [math.CO], 2020.
Miklós Bóna, Elijah DeJonge, Pattern Avoiding Permutations and Involutions with a Unique Longest Increasing Subsequence, (2020).


FORMULA

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.
a(n) = A152879(n,1).
a(n) = Sum_{k=1..n} k*A152879(n1,k).


EXAMPLE

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.


MAPLE

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);


CROSSREFS

Cf. A152879.
Sequence in context: A148775 A148776 A127385 * A259441 A176605 A080410
Adjacent sequences: A152877 A152878 A152879 * A152881 A152882 A152883


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Jan 02 2009


STATUS

approved



