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 A191313 Sum of the abscissae of the first returns to the horizontal axis (assumed to be 0 if there are no such returns) in all dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights). 2
 0, 0, 2, 5, 15, 30, 71, 134, 296, 551, 1188, 2211, 4720, 8815, 18722, 35105, 74307, 139842, 295223, 557366, 1174031, 2222606, 4672473, 8866776, 18607461, 35384676, 74139407, 141248270, 295524297, 563959752, 1178389423, 2252131246, 4700155088, 8995122383, 18751860084 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) = Sum_{k>=0} k*A191312(n,k). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 FORMULA G.f.: g = z*(4*z-1+q)/(q*(1-z)^2*(1-2*z+q)), where q=sqrt(1-4*z^2). a(n) ~ 2^n * (1 + 1/sqrt(2*Pi*n) + 1/3*(-1)^n/sqrt(2*Pi*n)). - Vaclav Kotesovec, Mar 20 2014 Conjecture: n*(3*n-13)*a(n) +2*(-6*n^2+29*n-18)*a(n-1) +(3*n^2-13*n+24)*a(n-2) +2*(21*n^2-124*n+150)*a(n-3) +4*(-15*n^2+92*n-132) *a(n-4) +8*(n-3)*(3*n-10) *a(n-5)=0. - R. J. Mathar, Jun 14 2016 EXAMPLE a(4)=15 because the sum of the abscissae of the first returns in HHHH, HHUD, HUDH, UDHH, UDUD, and UUDD is 0+4+3+2+2+4=15; here H=(1,0), U=(1,1), and D=(1,-1). MAPLE g := z*(4*z-1+sqrt(1-4*z^2))/((1-z)^2*sqrt(1-4*z^2)*(1-2*z+sqrt(1-4*z^2))): gser := series(g, z = 0, 37): seq(coeff(gser, z, n), n = 0 .. 34); MATHEMATICA CoefficientList[Series[x*(4*x-1+Sqrt[1-4*x^2])/((1-x)^2*Sqrt[1-4*x^2]*(1-2*x+Sqrt[1-4*x^2])), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *) CROSSREFS Cf. A191312. Partial sums of A226881. Sequence in context: A287843 A290828 A290836 * A078528 A309272 A077686 Adjacent sequences: A191310 A191311 A191312 * A191314 A191315 A191316 KEYWORD nonn AUTHOR Emeric Deutsch, May 30 2011 STATUS approved

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Last modified June 9 13:19 EDT 2023. Contains 363180 sequences. (Running on oeis4.)