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).

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

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