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 A001558 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) 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 Convolution of A000108 with A104629. - Philippe Deléham, Nov 11 2009 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 REFERENCES 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). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265. T. Fine, Extrapolation when very little is known about the source Information and Control 16 (1970), 331-359. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics. Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2. FORMULA a(n) = A000957(n+4) - A000957(n+3) - A000957(n+2) (A000957 are the Fine numbers). - Emeric Deutsch, May 08 2006 a(n) = A118972(n+3,1). - Emeric Deutsch, May 08 2006 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 a(n) = Sum_{k>=0} k*A111301(n+2,k). - Emeric Deutsch, Oct 05 2008 (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 a(n) ~ 11*2^(2*n+4)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014 EXAMPLE 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 + ... MAPLE 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 MATHEMATICA 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 *) PROG (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:=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 CROSSREFS Cf. A000957, A118972, A118973. Cf. A111301. - Emeric Deutsch, Oct 05 2008 Sequence in context: A257363 A071722 A058987 * A304824 A111639 A302076 Adjacent sequences:  A001555 A001556 A001557 * A001559 A001560 A001561 KEYWORD nonn,easy AUTHOR EXTENSIONS Edited by Emeric Deutsch, May 08 2006 STATUS approved

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Last modified September 15 20:35 EDT 2019. Contains 327087 sequences. (Running on oeis4.)