OFFSET
0,8
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{k=0..floor((n-3)/2)} k*A191518(n,k) for n>=4 (clarified by G. C. Greubel).
G.f.: (1-3*z^2-(1-z^2)*sqrt(1-4*z^2))/(2*(1-2*z)*sqrt(1-4*z^2)).
a(n) ~ 2^(n-5/2)*sqrt(n)/sqrt(Pi) * (1 - 3*sqrt(Pi)/sqrt(2*n)). - Vaclav Kotesovec, Mar 21 2014
D-finite with recurrence n*(n-6)*a(n) -2*n*(n-6)*a(n-1) -4*(n-3)*(n-4)*a(n-2) +8*(n-3)*(n-4)*a(n-3)=0. - R. J. Mathar, Jul 24 2022
EXAMPLE
a(7)=2 because among the 35 (=A001405(7)) dispersed Dyck paths of length 7 only UUUDDDH and HUUUDDD have UUU's.
MAPLE
g := ((1-3*z^2-(1-z^2)*sqrt(1-4*z^2))*1/2)/((1-2*z)*sqrt(1-4*z^2)): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);
MATHEMATICA
CoefficientList[Series[((1-3*x^2-(1-x^2)*Sqrt[1-4*x^2])*1/2)/((1-2*x)* Sqrt[1-4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *)
PROG
(PARI) x='x+O('x^50); concat([0, 0, 0, 0, 0, 0], Vec((1-3*x^2-(1-x^2)*sqrt(1-4*x^2))/(2*(1-2*x)*sqrt(1-4*x^2)))) \\ G. C. Greubel, Mar 26 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 07 2011
STATUS
approved