A191520 Number of UUU's in all the dispersed Dyck paths of semilength n (i.e., in all Motzkin paths of length n; U=(1,1)).

0, 0, 0, 0, 0, 0, 1, 2, 9, 18, 57, 114, 312, 624, 1578, 3156, 7599, 15198, 35401, 70802, 161052, 322104, 719790, 1439580, 3173090, 6346180, 13836426, 27672852, 59803104, 119606208, 256596276, 513192552, 1094249019, 2188498038, 4642178601, 9284357202

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

Cf. A001405, A191518.

Sequence in context: A200085 A083708 A280588 * A037421 A083423 A068978

Adjacent sequences: A191517 A191518 A191519 * A191521 A191522 A191523

Keyword

nonn

Author

Emeric Deutsch, Jun 07 2011

Status

approved