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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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Helmut Prodinger, Dispersed Dyck paths revisited, arXiv:2402.13026 [math.CO], 2024.
FORMULA
a(n) = A191387(n,0).
G.f.: (3-sqrt(1-4*z^2))/(2-3*z+z*sqrt(1-4*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..(n-1)/2}(Sum_{k=0..i}((k+1)*binomial(2*i-k,i-k)*binomial(n-2*i-1,k+1))/(i+1)). - Vladimir Kruchinin, Mar 27 2016
D-finite with recurrence -n*a(n) +3*n*a(n-1) +2*(n-6)*a(n-2) +12*(-n+3)*a(n-3) +(7*n-24)*a(n-4) +4*(n-3)*a(n-6)=0. - R. J. Mathar, Sep 24 2021
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 := (3-sqrt(1-4*z^2))/(2-3*z+z*sqrt(1-4*z^2)): gser := series(g, z = 0, 42): seq(coeff(gser, z, n), n = 0 .. 38);
MATHEMATICA
CoefficientList[Series[(3-Sqrt[1-4*x^2])/(2-3*x+x*Sqrt[1-4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 21 2014 *)
PROG
(Maxima)
a(n):=1+sum(sum((k+1)*binomial(2*i-k, i-k)*binomial(n-2*i-1, k+1), k, 0, i)/(i+1), i, 0, (n-1)/2); /* Vladimir Kruchinin, Mar 27 2016 */
(PARI) x='x+O('x^99); Vec((3-sqrt(1-4*x^2))/(2-3*x+x*sqrt(1-4*x^2))) \\ Altug Alkan, Mar 27 2016
CROSSREFS
Cf. A191387.
Sequence in context: A246360 A005627 A191794 * A194850 A062692 A182024
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 02 2011
STATUS
approved

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Last modified April 18 15:48 EDT 2024. Contains 371780 sequences. (Running on oeis4.)