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A001558
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Number of hill-free Dyck paths of semilength n+3 and having length of first descent equal to 1 (a hill in a Dyck path is a peak at level 1).
(Formerly M2845 N1143)
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7
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1, 3, 10, 33, 111, 379, 1312, 4596, 16266, 58082, 209010, 757259, 2760123, 10114131, 37239072, 137698584, 511140558, 1904038986, 7115422212, 26668376994, 100221202998, 377570383518, 1425706128480, 5394898197448, 20454676622476
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OFFSET
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0,2
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COMMENTS
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a(n) is also the number of even-length descents to ground level in all Dyck paths of semilength n+2. Example: a(1)=3 because in UDUDUD, UDUU(DD), UU(DD)UD, UUDU(DD) and UUUDDD we have 3 even-length descents to ground level (shown between parentheses). - Emeric Deutsch, Oct 05 2008
The Kn12 triangle sums of A039599 are given by the terms of this sequence. For the definition of this and other triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011
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REFERENCES
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: F*C^3, where F = (1-sqrt(1-4z))/(z*(3-sqrt(1-4z))) and C = (1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch, May 08 2006
(n+3)*a(n) = (-(11/2)*n + 21/2)*a(n-3) + ((9/2)*n + 11/2)*a(n-1) + (-(1/2)*n + 9/2)*a(n-2) + (-2n + 5)*a(n-4). - Simon Plouffe, Feb 09 2012
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EXAMPLE
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a(1)=3 because we have uu(d)ududd, uuu(d)uddd and uu(d)uuddd, where u=(1,1), d=(1,-1) (the first descents are shown between parentheses).
G.f. = 1 + 3*x + 10*x^2 + 33*x^3 + 111*x^4 + 379*x^5 + 1312*x^6 + ...
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MAPLE
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F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=F*C^3: gser:=series(g, z=0, 32): seq(coeff(gser, z, n), n=0..27); # Emeric Deutsch, May 08 2006
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MATHEMATICA
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CoefficientList[Series[(1-Sqrt[1-4*x])/(x*(3-Sqrt[1-4*x]))*((1-Sqrt[1-4*x])/(2*x))^3, {x, 0, 30}], x] (* Vaclav Kotesovec, Mar 20 2014 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec((1-sqrt(1-4*x))/(x*(3-sqrt(1-4*x)))*((1-sqrt(1-4*x))/(2*x))^3) \\ G. C. Greubel, Feb 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-Sqrt(1-4*x))/(x*(3-Sqrt(1-4*x)))*((1-Sqrt(1-4*x))/(2*x))^3 )); // G. C. Greubel, Feb 12 2019
(Sage) ((1-sqrt(1-4*x))/(x*(3-sqrt(1-4*x)))*((1-sqrt(1-4*x))/(2*x))^3).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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