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A191531 Sum of lengths of initial and final horizontal segments over all dispersed Dyck paths of semilength n (i.e., over all Motzkin paths of length n with no (1,0)-steps at positive heights). 2
0, 1, 2, 5, 10, 21, 40, 79, 148, 287, 538, 1041, 1964, 3811, 7242, 14105, 26974, 52713, 101332, 198571, 383326, 752837, 1458268, 2869131, 5573286, 10981597, 21382196, 42183395, 82299994, 162533193, 317650712, 627885751, 1228966140, 2431126919, 4764733138, 9431945577 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

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

FORMULA

a(n) = Sum_{k>=0} k*A191530(n,k).

G.f.: z*(3-2*z-sqrt(1-4*z^2))/((1-z)^2*(1-2*z+sqrt(1-4*z^2))).

a(n) ~ 2^(n+3/2)/sqrt(Pi*n). - Vaclav Kotesovec, Mar 21 2014

Conjecture: +n*(n-3)*a(n) -2*(n^2-2*n-2)*a(n-1) -(3*n^2-21*n+32)*a(n-2) +2*(n-2)*(4*n-13)*a(n-3) -4*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Jun 14 2016

EXAMPLE

a(4)=10 because in HHHH, HHUD, HUDH, UDHH, UDUD, UUDD we have 4+2+2+2+0+0=10.

MAPLE

g := z*(3-2*z-sqrt(1-4*z^2))/((1-z)^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[x (3-2x-Sqrt[1-4x^2])/((1-x)^2 (1-2x+ Sqrt[1-4x^2])) , {x, 0, 40}], x] (* Harvey P. Dale, Jun 19 2011 *)

PROG

(PARI) z='z+O('z^50); concat([0], Vec(z*(3-2*z-sqrt(1-4*z^2))/((1-z)^2*(1-2*z+sqrt(1-4*z^2))))) \\ G. C. Greubel, Mar 27 2017

CROSSREFS

Cf. A191530.

Sequence in context: A182807 A306098 A056599 * A212531 A261681 A030525

Adjacent sequences:  A191528 A191529 A191530 * A191532 A191533 A191534

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jun 07 2011

STATUS

approved

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Last modified August 20 06:36 EDT 2019. Contains 326139 sequences. (Running on oeis4.)