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A116914
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Number of UUDD's, where U=(1,1) and D=(1,-1), in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1).
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3
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1, 1, 5, 16, 58, 211, 781, 2920, 11006, 41746, 159154, 609324, 2341060, 9021559, 34855741, 134972368, 523689718, 2035462990, 7923732118, 30889008112, 120566373676, 471134916286, 1842964183570, 7216096752496, 28279240308268, 110913181145716, 435333520075796, 1709861650762900
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OFFSET
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2,3
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COMMENTS
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LINKS
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FORMULA
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a(n) = Sum_{k=0..floor(n/2)} k*A105640(n,k).
G.f.: x*(1 + 5*x - (1-x)*sqrt(1-4*x))/(2*(2+x)^2*sqrt(1-4*x)).
D-finite with recurrence +2*(-n+1)*a(n) +3*(-n+6)*a(n-1) +3*(13*n-44)*a(n-2) +10*(2*n-5)*a(n-3)=0. - R. J. Mathar, Jul 26 2022
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EXAMPLE
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a(4)=5 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UU(UUDD)DD, UUUDUDDD, UUD(UUDD)D, UUDUDUDD, U(UUDD)UDD and (UUDD)(UUDD) (U=(1,1), D=(1,-1)) we have altogether 5 UUDD's (shown between parentheses).
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MAPLE
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G:=z*(1+5*z-(1-z)*sqrt(1-4*z))/2/(2+z)^2/sqrt(1-4*z): Gser:=series(G, z=0, 31): seq(coeff(Gser, z^n), n=2..28);
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MATHEMATICA
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Rest[Rest[CoefficientList[Series[x*(1+5*x-(1-x)*Sqrt[1-4*x])/2/(2+x)^2/Sqrt[1-4*x], {x, 0, 40}], x]]] (* Vaclav Kotesovec, Mar 20 2014 *)
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PROG
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(PARI) my(x='x+O('x^40)); Vec(x*(1+5*x-(1-x)*sqrt(1-4*x))/(2*(2+x)^2 *sqrt(1-4*x))) \\ G. C. Greubel, Feb 08 2017
(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( x*(1 + 5*x-(1-x)*Sqrt(1-4*x))/(2*(2+x)^2*Sqrt(1-4*x)) )); // G. C. Greubel, May 08 2019
(Sage) a=(x*(1+5*x-(1-x)*sqrt(1-4*x))/(2*(2+x)^2*sqrt(1-4*x))).series(x, 40).coefficients(x, sparse=False); a[2:] # G. C. Greubel, May 08 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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