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 A129171 Sum of the heights of the peaks in all skew Dyck paths of semilength n. 2
 0, 1, 6, 32, 165, 840, 4251, 21443, 107946, 542680, 2725635, 13679997, 68623176, 344090307, 1724754180, 8642952000, 43300971885, 216895107480, 1086253033035, 5439405705125, 27234492215400, 136345625309965, 682531666024170 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1) (up), D=(1,-1) (down) and L=(-1,-1) (left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. LINKS G. C. Greubel and Vincenzo Librandi, Table of n, a(n) for n = 0..1000(terms 1..300 from Vincenzo Librandi) E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203. FORMULA a(n) = Sum_{k=0,..,n} k*A129170(n,k). G.f.: z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2. - corrected by Vaclav Kotesovec, Oct 20 2012 Recurrence: (n-1)*a(n) = (11*n-19)*a(n-1) - 5*(7*n-17)*a(n-2) + 25*(n-3)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012 a(n) ~ 3*5^(n-1)/2*(1-sqrt(5)/(6*sqrt(Pi*n))) . - Vaclav Kotesovec, Oct 20 2012 EXAMPLE a(2)=6 because in the 3 skew Dyck paths of semilength 2, namely UDUD, UUDD and UUDL, the heights of the peaks are 1,1,2 and 2. MAPLE G:=z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2: Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27); MATHEMATICA CoefficientList[Series[x*(3 - 3*x - Sqrt[1 - 6*x + 5*x^2])/(1 - 6*x + 5*x^2)/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *) PROG (PARI) z='z+O('z^25); concat(, Vec(z*(3-3*z-sqrt(1-6*z+5*z^2))/(1-6*z+5*z^2)/2)) \\ G. C. Greubel, Feb 10 2017 CROSSREFS Cf. A129170. Sequence in context: A097139 A034942 A046714 * A082585 A084326 A199699 Adjacent sequences:  A129168 A129169 A129170 * A129172 A129173 A129174 KEYWORD nonn AUTHOR Emeric Deutsch, Apr 07 2007 STATUS approved

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Last modified November 13 12:45 EST 2019. Contains 329094 sequences. (Running on oeis4.)