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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

%I #18 Jul 24 2022 13:18:41

%S 0,1,1,3,5,12,22,49,93,200,386,814,1586,3304,6476,13381,26333,54096,

%T 106762,218386,431910,880616,1744436,3547658,7036530,14281072,

%U 28354132,57451164,114159428,230993296,459312152,928319149,1846943453,3729244576,7423131482,14975907754

%N 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).

%F a(n) = Sum_{k>=0} k*A191390(n,k).

%F G.f.: g(z) = 4*z*(1-z)/(1-2*z+sqrt(1-4*z^2))^2.

%F 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]

%F 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

%e 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).

%p 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);

%Y Cf. A191390.

%Y First differences of A045621.

%K nonn

%O 0,4

%A _Emeric Deutsch_, Jun 03 2011

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Last modified April 16 03:28 EDT 2024. Contains 371696 sequences. (Running on oeis4.)