

A191388


Number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) with no valleys at level 0.


2



1, 1, 2, 3, 5, 8, 14, 23, 41, 69, 125, 214, 393, 682, 1267, 2223, 4171, 7385, 13976, 24935, 47544, 85377, 163863, 295900, 571216, 1036471, 2011130, 3664548, 7143068, 13063637, 25568085, 46912433, 92152906, 169570215, 334194418, 616530391, 1218694221, 2253451666, 4466410838
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OFFSET

0,3


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000


FORMULA

a(n) = A191387(n,0).
G.f.: (3sqrt(14*z^2))/(23*z+z*sqrt(14*z^2)).
a(n) ~ 2^(n+5/2) * (1+(1)^n/49) / (sqrt(Pi)*n^(3/2)).  Vaclav Kotesovec, Mar 21 2014
a(n) = 1+Sum_{i=0..(n1)/2}(Sum_{k=0..i}((k+1)*binomial(2*ik,ik)*binomial(n2*i1,k+1))/(i+1)).  Vladimir Kruchinin, Mar 27 2016
Conjecture: n*a(n) +(4*n+1)*a(n1) +(n+9)*a(n2) +2*(7*n25)*a(n3) +(19*n+72)*a(n4) +(7*n31)*a(n5) +4*(n+3)*a(n6) +4*(n4)*a(n7)=0.  R. J. Mathar, Jun 14 2016


EXAMPLE

a(4)=5 because we have HHHH, HHUD, HUDH, UDHH, and UUDD, where U=(1,1), H=(1,0), and D=(1,1) (UDUD does not qualify).


MAPLE

g := (3sqrt(14*z^2))/(23*z+z*sqrt(14*z^2)): gser := series(g, z = 0, 42): seq(coeff(gser, z, n), n = 0 .. 38);


MATHEMATICA

CoefficientList[Series[(3Sqrt[14*x^2])/(23*x+x*Sqrt[14*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *)


PROG

(Maxima)
a(n):=1+sum(sum((k+1)*binomial(2*ik, ik)*binomial(n2*i1, k+1), k, 0, i)/(i+1), i, 0, (n1)/2); /* Vladimir Kruchinin, Mar 27 2016 */
(PARI) x='x+O('x^99); Vec((3sqrt(14*x^2))/(23*x+x*sqrt(14*x^2))) \\ Altug Alkan, Mar 27 2016


CROSSREFS

Cf. A191387.
Sequence in context: A246360 A005627 A191794 * A194850 A062692 A182024
Adjacent sequences: A191385 A191386 A191387 * A191389 A191390 A191391


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Jun 02 2011


STATUS

approved



