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%I #23 Jul 15 2022 02:37:12
%S 0,0,2,5,15,30,71,134,296,551,1188,2211,4720,8815,18722,35105,74307,
%T 139842,295223,557366,1174031,2222606,4672473,8866776,18607461,
%U 35384676,74139407,141248270,295524297,563959752,1178389423,2252131246,4700155088,8995122383,18751860084
%N 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).
%C a(n) = Sum_{k>=0} k*A191312(n,k).
%H Vincenzo Librandi, <a href="/A191313/b191313.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: g = z*(4*z-1+q)/(q*(1-z)^2*(1-2*z+q)), where q=sqrt(1-4*z^2).
%F a(n) ~ 2^n * (1 + 1/sqrt(2*Pi*n) + 1/3*(-1)^n/sqrt(2*Pi*n)). - _Vaclav Kotesovec_, Mar 20 2014
%F 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
%e 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).
%p 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);
%t 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 *)
%Y Cf. A191312.
%Y Partial sums of A226881.
%K nonn
%O 0,3
%A _Emeric Deutsch_, May 30 2011
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