

A191391


Number of horizontal segments in all dispersed Dyck paths of length n (i.e., in all Motzkin paths of length n with no (1,0)steps at positive heights; a horizontal segment is a maximal sequence of consecutive (1,0)steps).


2



0, 1, 1, 3, 5, 12, 22, 49, 93, 200, 386, 814, 1586, 3304, 6476, 13381, 26333, 54096, 106762, 218386, 431910, 880616, 1744436, 3547658, 7036530, 14281072, 28354132, 57451164, 114159428, 230993296, 459312152, 928319149, 1846943453, 3729244576, 7423131482, 14975907754
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OFFSET

0,4


LINKS

Table of n, a(n) for n=0..35.


FORMULA

a(n) = Sum_{k>=0} k*A191390(n,k).
G.f.: g(z) = 4*z*(1z)/(12*z+sqrt(14*z^2))^2.
a(0)=0 and a(n)=2^(n1)C(n1,floor(n/2)1) for n>=1. [Joerg Arndt, Aug 07 2012, aeb]
Dfinite with recurrence (n+1)*a(n) +(3*n1)*a(n1) +2*(n+3)*a(n2) +4*(3*n8)*a(n3) +8*(n+4)*a(n4)=0.  R. J. Mathar, Jul 24 2022


EXAMPLE

a(4)=5 because in (HHHH), (HH)UD, (H)UD(H), UD(HH), UDUD, and UUDD we have a total of 1+1+2+1+0+0=5 horizontal segments (shown between parentheses).


MAPLE

g := 4*z*(1z)/(12*z+sqrt(14*z^2))^2: gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 35);


CROSSREFS

Cf. A191390.
First differences of A045621.
Sequence in context: A034763 A183921 A177143 * A121482 A013498 A161624
Adjacent sequences: A191388 A191389 A191390 * A191392 A191393 A191394


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Jun 03 2011


STATUS

approved



