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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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
FORMULA
a(n) = Sum_{k>=0} k*A191390(n,k).
G.f.: g(z) = 4*z*(1-z)/(1-2*z+sqrt(1-4*z^2))^2.
a(0)=0 and a(n)=2^(n-1)-C(n-1,floor(n/2)-1) for n>=1. [Joerg Arndt, Aug 07 2012, aeb]
D-finite with recurrence (n+1)*a(n) +(-3*n-1)*a(n-1) +2*(-n+3)*a(n-2) +4*(3*n-8)*a(n-3) +8*(-n+4)*a(n-4)=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*(1-z)/(1-2*z+sqrt(1-4*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 A368858 A013498
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 03 2011
STATUS
approved

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Last modified February 21 20:29 EST 2024. Contains 370237 sequences. (Running on oeis4.)