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).

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

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 A333798

Adjacent sequences: A191528 A191529 A191530 * A191532 A191533 A191534

Keyword

nonn

Author

Emeric Deutsch, Jun 07 2011

Status

approved