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 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Also number of peaks of maximum height in all Dyck paths of semilength n-1. Example: a(3)=3 because in (UD)(UD) and U(UD)D we have three peaks of maximum height (shown between parentheses). LINKS 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(j-1)-zf(j-2), j>=2. a(n) = A152879(n,1). a(n) = Sum_{k=1..n} k*A152879(n-1,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 := 1: f := 1: for i from 2 to 35 do f[i] := sort(expand(f[i-1]-z*f[i-2])) 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

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Last modified June 16 21:55 EDT 2021. Contains 345080 sequences. (Running on oeis4.)