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A152880
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Number of Dyck paths of semilength n having exactly one peak of maximum height.
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5
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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, 35406650450001, 134215036793130
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OFFSET
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1,3
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COMMENTS
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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).
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LINKS
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FORMULA
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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) = Sum_{k=1..n} k*A152879(n-1,k).
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EXAMPLE
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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.
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MAPLE
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f[0] := 1: f[1] := 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);
# second Maple program:
b:= proc(x, y, h, c) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, c, add(b(x-1, y-i, max(h, y), `if`(h=y, 0,
`if`(h<y, 1, c))), i=[1, -1])))
end:
a:= n-> b(2*n, 0$3):
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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