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
Keywords
Examples
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).
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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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);
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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 *)
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
Comments